INTERNATIONAL GONGRESS lass="math-inline "> { μ p n r } n μ p n ⊗ r n {mu_(p^(n))^(ox r)}_(n)\left\{\mu_{p^{n}}^{\otimes r}\right\}_{n}{μpn⊗r}n on the rigid analytic generic fiber X X XXX of X X X\mathcal{X}X and R ψ R ψ R psiR \psiRψ is the nearby cycles functor.
The isomorphism in (1) above, combined with the main result of [52], gives us the identification of pro-objects
Z p B M S ( r ) X { Z / p n ( r ) X , ét } n Z p B M S ( r ) X ≅ Z / p n ( r ) X ,  ét  n Z_(p)^(BMS)(r)_(X)~={Z//p^(n)(r)_(X," ét ")}_(n)\mathbb{Z}_{p}^{\mathrm{BMS}}(r)_{X} \cong\left\{\mathbb{Z} / p^{n}(r)_{X, \text { ét }}\right\}_{n}ZpBMS(r)X≅{Z/pn(r)X, ét }n
in the setting of (1).
Consider the case of a smooth O K O K O_(K)\mathcal{O}_{K}OK-scheme X X XXX as before. Bhatt-Morrow-Scholze suggest in [18, REMARK 1.16] that Z p B M S / p n ( r ) X Z p B M S / p n ( r ) X Z_(p)^(BMS)//p^(n)(r)_(X)\mathbb{Z}_{p}^{\mathrm{BMS}} / p^{n}(r)_{X}ZpBMS/pn(r)X should be Schneider's sheaf S n ( r ) S n ( r ) S_(n)(r)S_{n}(r)Sn(r), and by passage to the limit, there should be a distinguished triangle
(6.6) i W Ω log r 1 [ r 1 ] Z p B M S ( r ) τ r R j ( Z p ( r ) V , e t ) i W Ω log r 1 [ r ] (6.6) i ∗ W Ω log r − 1 [ − r − 1 ] → Z p B M S ( r ) → Ï„ ≤ r R j ∗ Z p ( r ) V , e t → i ∗ W Ω log r − 1 [ − r ] {:(6.6)i_(**)WOmega_(log)^(r-1)[-r-1]rarrZ_(p)^(BMS)(r)rarrtau_( <= r)Rj_(**)(Z_(p)(r)_(V,et))rarri_(**)WOmega_(log)^(r-1)[-r]:}\begin{equation*} i_{*} W \Omega_{\log }^{r-1}[-r-1] \rightarrow \mathbb{Z}_{p}^{\mathrm{BMS}}(r) \rightarrow \tau_{\leq r} R j_{*}\left(\mathbb{Z}_{p}(r)_{V, \mathrm{et}}\right) \rightarrow i_{*} W \Omega_{\log }^{r-1}[-r] \tag{6.6} \end{equation*}(6.6)i∗WΩlogr−1[−r−1]→ZpBMS(r)→τ≤rRj∗(Zp(r)V,et)→i∗WΩlogr−1[−r]
For X X XXX a smooth O K O K O_(K)\mathcal{O}_{K}OK-scheme with associated formal scheme X X X\mathcal{X}X and special fiber i : Y X i : Y → X i:Y rarrXi: Y \rightarrow \mathcal{X}i:Y→X, this would give an isomorphism of i Z p B M S / p n ( r ) X i ∗ Z p B M S / p n ( r ) X i^(**)Z_(p)^(BMS)//p^(n)(r)Xi^{*} \mathbb{Z}_{p}^{\mathrm{BMS}} / p^{n}(r) Xi∗ZpBMS/pn(r)X with the étale motivic complex i Z / p n ( r ) ét i ∗ Z / p n ( r ) ét  i^(**)Z//p^(n)(r)_("ét ")i^{*} \mathbb{Z} / p^{n}(r)_{\text {ét }}i∗Z/pn(r)ét  on Y ét Y ét  Y_("ét ")Y_{\text {ét }}Yét  considered by Geisser.
This has been proven in a work-in-progress by Bhargav Bhatt and Akhil Mathew [16]. They construct an isomorphism of a version of Z p B M S / p n ( r ) X Z p B M S / p n ( r ) X Z_(p)^(BMS)//p^(n)(r)_(X)\mathbb{Z}_{p}^{\mathrm{BMS}} / p^{n}(r)_{X}ZpBMS/pn(r)X with Sato's sheaf T n ( r ) X T n ( r ) X T_(n)(r)_(X)\mathfrak{T}_{n}(r)_{X}Tn(r)X in the semi-stable case; using Zhong's extension of Geisser's results, this gives an isomorphism
i Z p B M S / p n ( r ) x i τ r Z / p n ( r ) êt i ∗ Z p B M S / p n ( r ) x ≅ i ∗ Ï„ ≤ r Z / p n ( r ) êt  i^(**)Z_(p)^(BMS)//p^(n)(r)x~=i^(**)tau_( <= r)Z//p^(n)(r)_("êt ")i^{*} \mathbb{Z}_{p}^{\mathrm{BMS}} / p^{n}(r) x \cong i^{*} \tau_{\leq r} \mathbb{Z} / p^{n}(r)_{\text {êt }}i∗ZpBMS/pn(r)x≅i∗τ≤rZ/pn(r)êt 
in the semi-stable case.
One has the Geisser-Hesselholt isomorphism (Theorem 6.8) of étale K K KKK-theory and topological cyclic homology given by the cyclotomic trace map. Perhaps one can compare the localization pro-distinguished triangle
K ( Y ; Z p ) K ( X ; Z p ) K ( X Y ; Z p ) K Y ; Z p → K X ; Z p → K X ∖ Y ; Z p K(Y;Z_(p))rarr K(X;Z_(p))rarr K(X\\Y;Z_(p))K\left(Y ; \mathbb{Z}_{p}\right) \rightarrow K\left(X ; \mathbb{Z}_{p}\right) \rightarrow K\left(X \backslash Y ; \mathbb{Z}_{p}\right)K(Y;Zp)→K(X;Zp)→K(X∖Y;Zp)
with the distinguished triangle (6.6). Assuming one does have the pro-isomorphism Z p ( r ) et Z p B M S ( r ) Z p ( r ) et  ≅ Z p B M S ( r ) Z_(p)(r)_("et ")~=Z_(p)^(BMS)(r)\mathbb{Z}_{p}(r)_{\text {et }} \cong \mathbb{Z}_{p}^{\mathrm{BMS}}(r)Zp(r)et ≅ZpBMS(r) as suggested above, it would be interesting to see if the identification of the sheaves S n ( r ) S n ( r ) S_(n)(r)S_{n}(r)Sn(r) with the étale motivic complexes Z / p n ( r ) ét Z / p n ( r ) ét  Z//p^(n)(r)_("ét ")\mathbb{Z} / p^{n}(r)_{\text {ét }}Z/pn(r)ét  and the Atiyah-Hirzebruch spectral sequence from motivic cohomology to K K KKK-theory could yield a comparison with the spectral sequence corresponding to the motivic tower Fil T C ( X ; Z p ) ∗ T C X ; Z p ^(**)TC(X;Z_(p)){ }^{*} \mathrm{TC}\left(\mathcal{X} ; \mathbb{Z}_{p}\right)∗TC(X;Zp) described above.
The sheaf Z p B M S ( r ) Z p B M S ( r ) Z_(p)^(BMS)(r)\mathbb{Z}_{p}^{\mathrm{BMS}}(r)ZpBMS(r) is built from T C ( ; Z p ) T C − ; Z p TC(-;Z_(p))\mathrm{TC}\left(-; \mathbb{Z}_{p}\right)TC(−;Zp), which by the Geisser-Hesselholt theorem is p p ppp-completed étale K K KKK-theory. As we mentioned before, the Geisser-Hesselholt isomorphism arises at least in part from McCarthy's theorem identifying the relative K K KKK-theory and relative TC of the nilpotent thickenings X / ( π n ) X / Ï€ n X//(pi^(n))X /\left(\pi^{n}\right)X/(Ï€n) of the special fiber Y Y YYY. However, the motivic cohomology complexes do not detect the difference between X / ( π n ) X / Ï€ n X//(pi^(n))X /\left(\pi^{n}\right)X/(Ï€n) and Y Y YYY. Supposing again that one does have a pro-isomorphism Z p ( r ) ét Z p B M S ( r ) Z p ( r ) ét  ≅ Z p B M S ( r ) Z_(p)(r)_("ét ")~=Z_(p)^(BMS)(r)\mathbb{Z}_{p}(r)_{\text {ét }} \cong \mathbb{Z}_{p}^{\mathrm{BMS}}(r)Zp(r)ét ≅ZpBMS(r), this says that in mixed characteristic ( 0 , p ) ( 0 , p ) (0,p)(0, p)(0,p), one can still see the K K KKK-theory of the thickened fibers X / ( π n ) X / Ï€ n X//(pi^(n))X /\left(\pi^{n}\right)X/(Ï€n) reflected in the motivic complexes Z p ( r ) êt Z p ( r ) êt  Z_(p)(r)_("êt ")\mathbb{Z}_{p}(r)_{\text {êt }}Zp(r)êt .
I am not aware of a categorical framework for the tower Fil n T C ( A ~ ; Z p ) Fil n ⁡ T C A ~ ; Z p Fil^(n)TC(( tilde(A));Z_(p))\operatorname{Fil}^{n} \mathrm{TC}\left(\tilde{A} ; \mathbb{Z}_{p}\right)Filn⁡TC(A~;Zp) and its layers Z p B M S ( r ) Z p B M S ( r ) Z_(p)^(BMS)(r)\mathbb{Z}_{p}^{\mathrm{BMS}}(r)ZpBMS(r), analogous to the framework for Voevodsky's slice tower for K K KKK-theory given by S H ( k ) S H ( k ) SH(k)\mathrm{SH}(k)SH(k). As A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy invariance fails for these theories, one would need a stable homotopy theory with a weaker invariance property, perhaps modeled on the one of the categories of motives with modulus discussed in the previous section, for these theories to find a home, in which the Bhatt-Morrow-Scholze tower (6.5) would be seen as a parallel to Voevodsky's slice tower.

FUNDING

The author gratefully acknowledges support from the DFG through the SPP 1786, and through a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 832833).

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MARC LEVINE

Universität Duisburg-Essen, Fakultät Mathematik, Campus Essen, 45117 Essen, Germany, marc.levine@uni-due.de

4. ALGEBRAIC AND COMPLEX GEOMETRY

HOMOLOGICAL KNOT INVARIANTS FROM MIRROR SYMMETRY

MINA AGANAGIC

ABSTRACT

In 1999, Khovanov showed that a link invariant known as the Jones polynomial is the Euler characteristic of a homology theory. The knot categorification problem is to find a general construction of knot homology groups, and to explain their meaning - what are they homologies of?
Homological mirror symmetry, formulated by Kontsevich in 1994, naturally produces hosts of homological invariants. Typically though, it leads to invariants which have no particular interest outside of the problem at hand.
I showed recently that there is a new family of mirror pairs of manifolds, for which homological mirror symmetry does lead to interesting invariants and solves the knot categorification problem. The resulting invariants are computable explicitly for any simple Lie algebra, and certain Lie superalgebras.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 57K18; Secondary 14J33, 53D37, 53D45

KEYWORDS

Homological mirror symmetry, knot homology theory, categorification

1. INTRODUCTION

There are many beautiful strands that connect mathematics and physics. Two of the most fruitful ones are knot theory and mirror symmetry. I will describe a new connection between them. We will find a solution to the knot categorification problem as a new application of homological mirror symmetry.
In 1984, Jones constructed a polynomial invariant of links in R 3 R 3 R^(3)\mathbb{R}^{3}R3 [42]. The Jones polynomial is defined by picking a projection of the link to a plane, the skein relation it satisfies
q n / 2 P q n / 2 P χ = ( q 1 / 2 q 1 / 2 ) P ζ q n / 2 P 天  − q − n / 2 P χ ⋆ = q 1 / 2 − q − 1 / 2 P ζ ä¹… q^(n//2)P_("天 ")-q^(-n//2)P_(chi^(***))=(q^(1//2)-q^(-1//2))P_(zetaä¹…)\mathfrak{q}^{n / 2} P_{\text {天 }}-\mathfrak{q}^{-n / 2} P_{\chi^{\star}}=\left(\mathfrak{q}^{1 / 2}-\mathfrak{q}^{-1 / 2}\right) P_{\zeta ä¹…}qn/2P天 −q−n/2Pχ⋆=(q1/2−q−1/2)Pζ久
where n = 2 n = 2 n=2n=2n=2, and the value for the unknot. It has the same flavor as the Alexander polynomial, dating back to 1928 [8], which one gets by setting n = 0 n = 0 n=0n=0n=0 instead.
The proper framework for these invariants was provided by Witten in 1988, who showed that they originate from three-dimensional Chern-Simons theory based on a Lie algebra L g L g ^(L)g{ }^{L} \mathrm{~g}L g [82]. In particular, the Jones polynomial comes from L g = s u 2 L g = s u 2 ^(L)g=su_(2){ }^{L} \mathrm{~g}=\mathfrak{s u}_{2}L g=su2 with links colored by the defining two-dimensional representation. The Alexander polynomial comes from the same setting by taking L g L g ^(L)g{ }^{L} \mathrm{~g}L g to be a Lie superalgebra g l l 1 1 g l l 1 ∣ 1 gll_(1∣1)\mathfrak{g l} \mathfrak{l}_{1 \mid 1}gll1∣1. The resulting link invariants are known as the U q ( L g ) U q L g U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(L g) quantum group invariants. The relation to quantum groups was discovered by Reshetikhin and Turaev [67].

1.2. The knot categorification problem

The quantum invariants of links are Laurant polynomials in q 1 / 2 q 1 / 2 q^(1//2)q^{1 / 2}q1/2, with integer coefficients. In 1999, Khovanov showed [48, 49] that one can associate to a projection of the link to a plane a bigraded complex of vector spaces
C , j ( K ) = C i 1 , j ( K ) i 1 C i , j ( K ) i C ∗ , j ( K ) = ⋯ C i − 1 , j ( K ) → ∂ i − 1 C i , j ( K ) → ∂ i ⋯ C^(**,j)(K)=cdotsC^(i-1,j)(K)rarr"del^(i-1)"C^(i,j)(K)rarr"del^(i)"cdotsC^{*, j}(K)=\cdots C^{i-1, j}(K) \xrightarrow{\partial^{i-1}} C^{i, j}(K) \xrightarrow{\partial^{i}} \cdotsC∗,j(K)=⋯Ci−1,j(K)→∂i−1Ci,j(K)→∂i⋯
whose cohomology H i , j ( K ) = ker i / im i 1 H i , j ( K ) = ker ⁡ ∂ i / im ⁡ ∂ i − 1 H^(i,j)(K)=ker del^(i)//im del^(i-1)\mathscr{H}^{i, j}(K)=\operatorname{ker} \partial^{i} / \operatorname{im} \partial^{i-1}Hi,j(K)=ker⁡∂i/im⁡∂i−1 categorifies the Jones polynomial,
J K ( q ) = i , j ( 1 ) i q j / 2 r k H i , j ( K ) J K ( q ) = ∑ i , j   ( − 1 ) i q j / 2 r k H i , j ( K ) J_(K)(q)=sum_(i,j)(-1)^(i)q^(j//2)rkH^(i,j)(K)J_{K}(\mathfrak{q})=\sum_{i, j}(-1)^{i} \mathfrak{q}^{j / 2} \mathrm{rk} \mathscr{H}^{i, j}(K)JK(q)=∑i,j(−1)iqj/2rkHi,j(K)
Moreover, the cohomology groups
H , ( K ) = i , j H i , j ( K ) H ∗ , ∗ ( K ) = ⨁ i , j   H i , j ( K ) H^(**,**)(K)=bigoplus_(i,j)H^(i,j)(K)\mathscr{H}^{*, *}(K)=\bigoplus_{i, j} \mathscr{H}^{i, j}(K)H∗,∗(K)=⨁i,jHi,j(K)
are independent of the choice of projection; they are themselves link invariants.

1.2.1.

Khovanov's construction is part of the categorification program initiated by Crane and Frenkel [25], which aims to lift integers to vector spaces and vector spaces to categories.
A toy model of categorification comes from a Riemannian manifold M M MMM, whose Euler characteristic
χ ( M ) = k Z ( 1 ) k dim H k ( M ) χ ( M ) = ∑ k ∈ Z   ( − 1 ) k dim ⁡ H k ( M ) chi(M)=sum_(k inZ)(-1)^(k)dim H^(k)(M)\chi(M)=\sum_{k \in \mathbb{Z}}(-1)^{k} \operatorname{dim} \mathscr{H}^{k}(M)χ(M)=∑k∈Z(−1)kdim⁡Hk(M)
is categorified by the cohomology H k ( M ) = ker d k / im d k 1 H k ( M ) = ker ⁡ d k / im ⁡ d k − 1 H^(k)(M)=ker d_(k)//im d_(k-1)\mathscr{H}^{k}(M)=\operatorname{ker} d_{k} / \operatorname{im} d_{k-1}Hk(M)=ker⁡dk/im⁡dk−1 of the de Rham complex
C = C k 1 d k 1 C k d k . C ∗ = ⋯ C k − 1 → d k − 1 C k → d k ⋯ . C^(**)=cdotsC^(k-1)rarr"d_(k-1)"C^(k)rarr"d_(k)"cdots.C^{*}=\cdots C^{k-1} \xrightarrow{d_{k-1}} C^{k} \xrightarrow{d_{k}} \cdots .C∗=⋯Ck−1→dk−1Ck→dk⋯.
The Euler characteristic is, from the physics perspective, the partition function of supersymmetric quantum mechanics with M M MMM as a target space χ ( M ) = Tr ( 1 ) F e β H χ ( M ) = Tr ⁡ ( − 1 ) F e − β H chi(M)=Tr(-1)^(F)e^(-beta H)\chi(M)=\operatorname{Tr}(-1)^{F} e^{-\beta H}χ(M)=Tr⁡(−1)Fe−βH, with Laplacian H = d d + d d H = d d ∗ + d ∗ d H=dd^(**)+d^(**)dH=d d^{*}+d^{*} dH=dd∗+d∗d as the Hamiltonian, and d = k d k d = ∑ k   d k d=sum_(k)d_(k)d=\sum_{k} d_{k}d=∑kdk as the supersymmetry operator. If h h hhh is a Morse function on M M MMM, the complex can be replaced by a Morse-Smale-Witten complex C h C h ∗ C_(h)^(**)C_{h}^{*}Ch∗ with the differential d h = e h d e h d h = e h d e − h d_(h)=e^(h)de^(-h)d_{h}=e^{h} d e^{-h}dh=ehde−h. The complex C h C h ∗ C_(h)^(**)C_{h}^{*}Ch∗ is the space of perturbative ground states of a σ σ sigma\sigmaσ-model on M M MMM with potential h h hhh [81]. The action of the differential d h d h d_(h)d_{h}dh is generated by solutions to flow equations, called instantons.

1.2.2.

Khovanov's remarkable categorification of the Jones polynomial is explicit and easily computable. It has generalizations of similar flavor for L g = u n L g = u n ^(L)g=u_(n){ }^{L} \mathfrak{g}=\mathfrak{} u_{n}Lg=un, and links colored by its minuscule representations [51].
In 2013, Webster showed [78] that for any L g L g ^(L)g{ }^{L} \mathrm{~g}L g, there exists an algebraic framework for categorification of U q ( L g ) U q L g U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(L g) invariants of links in R 3 R 3 R^(3)\mathbb{R}^{3}R3, based on a derived category of modules of an associative algebra. The KLRW algebra, defined in [78], generalizes the algebras of Khovanov and Lauda [50] and Rouquier [68]. Unlike Khovanov's construction, Webster's categorification is anything but explicit.

1.2.3.

Despite the successes of the program, one is missing a fundamental principle which explains why is categorification possible - the construction has no right to exist. Unlike in our toy example of categorification of the Euler characteristic of a Riemanniann manifold, Khovanov's construction and its generalizations did not come from either geometry or physics in any unified way. The problem Khovanov initiated is to find a general framework for link homology, that works uniformly for all Lie algebras, explains what link homology groups are, and why they exist.

1.3. Homological invariants from mirror symmetry

The solution to the problem comes from a new relation between mirror symmetry and representation theory.
Homological mirror symmetry relates pairs of categories of geometric origin [55]: a derived category of coherent sheaves and a version of the derived Fukaya category, in which complementary aspects of the theory are simple to understand. Occasionally, one can make mirror symmetry manifest, by showing that both categories are equivalent to a derived category of modules of a single algebra.
I will describe a new family of mirror pairs, in which homological mirror symmetry can be made manifest and leads to the solution to the knot categorification problem [1, 2]. Many special features exist in this family, in part due to its deep connections to representation theory. As a result, the theory is solvable explicitly, as opposed to only formally [4,5].

1.4. The solution

We will get not one, but two solutions to the knot categorification problem. The first solution [1] is based on D X D X DX\mathscr{D} XDX, the derived category of equivariant coherent sheaves on a certain holomorphic symplectic manifold X X X\mathcal{X}X, which plays a role in the geometric Langlands correspondence. Recently, Webster proved that D X D X D_(X)\mathscr{D}_{X}DX is equivalent to D A D A D_(A)\mathscr{D}_{\mathscr{A}}DA, the derived category of modules of an algebra A A A\mathscr{A}A which is a cylindrical version of the KLRW algebra from [79,80]. The generalization allows the theory to describe links in R 2 × S 1 R 2 × S 1 R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}R2×S1, as well as in R 3 R 3 R^(3)\mathbb{R}^{3}R3.
The second solution [2] is based on D Y D Y D_(Y)\mathscr{D}_{Y}DY, the derived Fukaya-Seidel category of a certain manifold Y Y YYY with potential W W WWW. The theory generalizes Heegard-Floer theory [ 63 , 64 [ 63 , 64 [63,64[63,64[63,64, 66], which categorifies the Alexander polynomial, from L g = g l 1 1 L g = g l 1 ∣ 1 ^(L)g=gl_(1∣1){ }^{L} \mathrm{~g}=\mathrm{gl}_{1 \mid 1}L g=gl1∣1, to arbitrary L g L g ^(L)g{ }^{L} \mathrm{~g}L g.
The two solutions are related by equivariant homological mirror symmetry, which is not an equivalence of categories, but a correspondence of objects and morphisms coming from a pair of adjoint functors. In D X D X DX\mathscr{D} XDX, we will learn which question we need to ask to obtain U q ( L g ) U q L g U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(L g) link homology. In D Y D Y D_(Y)\mathscr{D}_{Y}DY, we will learn how to answer it.
In [5], we give an explicit algorithm for computing homological link invariants from D Y D Y D_(Y)\mathscr{D}_{Y}DY, for any simple Lie algebra L g L g ^(L)g{ }^{L} \mathrm{~g}L g and links colored by its minuscule representations. It has an extension to Lie superalgebras L g = g l m n L g = g l m ∣ n ^(L)g=gl_(m∣n){ }^{L} \mathfrak{g}=\mathfrak{g} \mathfrak{l}_{m \mid n}Lg=glm∣n and s p m 2 n s p m ∣ 2 n sp_(m∣2n)\mathfrak{s} \mathfrak{p}_{m \mid 2 n}spm∣2n. In [4], we set the mathematical foundations of D Y D Y D_(Y)\mathscr{D}_{Y}DY and prove (equivariant) homological mirror symmetry relating it to D X D X DX\mathscr{D} XDX.

2. KNOT INVARIANTS AND CONFORMAL FIELD THEORY

Most approaches to categorification of U q ( L g ) U q L g U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(L g) link invariants start with quantum groups and their modules. We will start by recalling how quantum groups came into the story. The seeming detour will help us understand how U q ( L g ) U q L g U_(q)(^(L)g)U_{\mathfrak{q}}\left({ }^{L} \mathfrak{g}\right)Uq(Lg) link invariants arise from geometry, and what categorifies them.

2.1. Knizhnik-Zamolodchikov equation and quantum groups

Chern-Simons theory associates to a punctured Riemann surface A A A\mathscr{A}A a vector space, its Hilbert space. As Witten showed [82], the Hilbert space is finite dimensional, and spanned by vectors that have a name. They are known as conformal blocks of the affine Lie algebra L ^ g κ L ^ g κ widehat(L)_(g_(kappa))\widehat{L}_{\mathrm{g}_{\kappa}}L^gκ. The effective level κ κ kappa\kappaκ is an arbitrary complex number, related to q q qqq by q = e 2 π i κ q = e 2 Ï€ i κ q=e^((2pi i)/(kappa))q=e^{\frac{2 \pi i}{\kappa}}q=e2Ï€iκ. While in principle arbitrary representations of L g L g ^(L)g{ }^{L} \mathrm{~g}L g can occur, in relating to geometry and categorification, we will take them to be minuscule.
To get invariants of knots in R 3 R 3 R^(3)\mathbb{R}^{3}R3, one typically takes A A A\mathscr{A}A to be a complex plane with punctures. It is equivalent, but for our purposes better, to take A A A\mathscr{A}A to be an infinite complex cylinder. This way, we will be able to describe invariants of links in R 2 × S 1 R 2 × S 1 R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}R2×S1, as well.

2.1.1.

Every conformal block, and hence every state in the Hilbert space, can be obtained explicitly as a solution to a linear differential equation discovered by Knizhnik and Zamolodchikov in 1984 [53]. The KZ equation we get is of trigonometric type, schematically
(2.1) κ i V = j i r i j ( a i / a j ) V (2.1) κ ∂ i V = ∑ j ≠ i   r i j a i / a j V {:(2.1)kappadel_(i)V=sum_(j!=i)r_(ij)(a_(i)//a_(j))V:}\begin{equation*} \kappa \partial_{i} \mathcal{V}=\sum_{j \neq i} r_{i j}\left(a_{i} / a_{j}\right) \mathcal{V} \tag{2.1} \end{equation*}(2.1)κ∂iV=∑j≠irij(ai/aj)V
since A A A\mathscr{A}A is an infinite cylinder. Here, i = a i a i ∂ i = a i ∂ ∂ a i del_(i)=a_(i)(del)/(dela_(i))\partial_{i}=a_{i} \frac{\partial}{\partial a_{i}}∂i=ai∂∂ai, where a i a i a_(i)a_{i}ai is any of the n n nnn punctures in the interior of A A A\mathcal{A}A, colored by a representation V i V i V_(i)V_{i}Vi of L g L g ^(L)g{ }^{L} \mathrm{~g}L g. The right hand side of (2.1) is given in terms of classical r r rrr-matrices of L g L g ^(L)g{ }^{L} \mathrm{~g}L g, and acts irreducibly on a subspace of V 1 V n V 1 ⊗ ⋯ ⊗ V n V_(1)ox cdots oxV_(n)V_{1} \otimes \cdots \otimes V_{n}V1⊗⋯⊗Vn of a fixed weight ν ν nu\nuν, where V V V\mathcal{V}V takes values [32,33].
The K Z K Z KZ\mathrm{KZ}KZ equations define a flat connection on a vector bundle over the configuration space of distinct points a 1 , , a n A a 1 , … , a n ∈ A a_(1),dots,a_(n)inAa_{1}, \ldots, a_{n} \in \mathcal{A}a1,…,an∈A. The flatness of the connection is the integrability condition for the equation.

2.1.2.

The monodromy problem of the K Z K Z KZ\mathrm{KZ}KZ equation, which is to describe analytic continuation of its fundamental solution along a path in the configuration space, has an explicit solution. Drinfeld [30] and Kazhdan and Lustig [47] proved that that the monodromy matrix B B B\mathfrak{B}B of the K Z K Z KZ\mathrm{KZ}KZ connection is a product of R R RRR-matrices of the U q ( L g ) U q L g U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(L g) quantum group corresponding to L g L g ^(L)g{ }^{L} \mathrm{~g}L g. The R R RRR-matrices describe reorderings of neighboring pairs of punctures.
The monodromy matrix B B B\mathfrak{B}B is the Chern-Simons path integral on A × [ 0 , 1 ] A × [ 0 , 1 ] Axx[0,1]\mathscr{A} \times[0,1]A×[0,1] in presence of a colored braid. By composing braids, we get a representation of the affine braid group based on the U q ( L g ) U q L g U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(L g) quantum group, acting on the space of solutions to the K Z K Z KZ\mathrm{KZ}KZ equation. The braid group is affine, since A A A\mathscr{A}A is a cylinder and not a plane.

2.1.3.

Any link can be represented as a plat closure of some braid. The Chern-Simons path integral together with the link computes a very specific matrix element of the braiding matrix B B B\mathfrak{B}B, picked out by a pair of states in the Hilbert space corresponding to the top and the bottom of Figure 1 .

FIGURE 1

Every link arises as a plat closure of a braid.
These states, describing a collection of cups and caps, are very special solutions of the K Z K Z KZ\mathrm{KZ}KZ equation in which pairs of punctures, colored by conjugate representations V i V i V_(i)V_{i}Vi and V i V i ∗ V_(i)^(**)V_{i}^{*}Vi∗, come together and fuse to disappear. In this way, both fusion and braiding enter the problem.

2.2. A categorification wishlist

To categorify U q ( L g ) U q L g U_(q)(^(L)(g))U_{\mathfrak{q}}\left({ }^{L} \mathrm{~g}\right)Uq(L g) invariants of links in R 3 R 3 R^(3)\mathbb{R}^{3}R3, we would like to associate, to the space of conformal blocks of L G ^ L G ^ widehat(L_(G))\widehat{L_{\mathcal{G}}}LG^ on the Riemann surface A A A\mathcal{A}A, a bigraded category, which in addition to the cohomological grading has a grading associated to q q q\mathrm{q}q. Additional r k ( L g ) r k L g rk(^(L)(g))\mathrm{rk}\left({ }^{L} \mathrm{~g}\right)rk(L g) gradings are needed to categorify invariants of links in R 2 × S 1 R 2 × S 1 R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}R2×S1, as they depend on the choice of a flat L g L g ^(L)g{ }^{L} \mathrm{~g}L g connection around the S 1 S 1 S^(1)S^{1}S1. To braids, we would like to associate functors between the categories corresponding to the top and bottom. To links, we would like to associate a vector space whose elements are morphisms between the objects of the categories associated to the top and bottom, up to the action of the braiding functor. Moreover, we would like to do this in a way that recovers quantum link invariants upon decategorification. One typically proceeds by coming up with a category, and then works to prove that decategorification gives the link invariants one set out to categorify. A virtue of the solutions in [ 1 , 2 ] [ 1 , 2 ] [1,2][1,2][1,2] is that the second step is automatic.

3. MIRROR SYMMETRY

Mirror symmetry is a string duality which relates σ σ sigma\sigmaσ-models on a pair of Calabi-Yau manifolds X X X\mathcal{X}X and y y y\mathscr{y}y. Its mathematical imprint are relations between very different problems in complex geometry of X X X\mathcal{X}X ("B-type") and symplectic geometry of y y yyy ("A-type"), and vice versa.
Mirror symmetry was discovered as a duality of σ σ sigma\sigmaσ-models on closed Riemann surfaces D D DDD. In string theory, one must allow Riemann surfaces with boundaries. This enriches the theory by introducing "branes," which are boundary conditions at D ∂ D del D\partial D∂D and naturally objects of a category [9].
By asking how mirror symmetry acts on branes turned out to yield deep insights into mirror symmetry. One such insight is due to Strominger, Yau, and Zaslov [75], who showed that in order for every point-like brane on X X X\mathcal{X}X to have a mirror on Y Y YYY, mirror manifolds have to be fibered by a pair of (special Lagrangian) dual tori T T TTT and T T ∨ T^(vv)T^{\vee}T∨, over a common base.

3.1. Homological mirror symmetry

Kontsevich conjectured in his 1994 ICM address [55] that mirror symmetry should be understood as an equivalence of a pair of categories of branes, one associated to complex geometry of X X X\mathcal{X}X, the other to symplectic geometry of y y y\mathscr{y}y.
The category of branes associated to complex geometry of X X X\mathcal{X}X is the derived category of coherent sheaves,
D X = D b Coh T ( X ) D X = D b Coh T ⁡ ( X ) D_(X)=D^(b)Coh_(T)(X)\mathscr{D}_{X}=D^{b} \operatorname{Coh}_{T}(\mathcal{X})DX=DbCohT⁡(X)
Its objects are "B-type branes," supported on complex submanifolds of X X X\mathcal{X}X. The category of branes associated to symplectic geometry is the derived Fukaya category
D y = D Fuk ( y ) D y = D Fuk ⁡ ( y ) Dy=D Fuk(y)\mathscr{D} y=D \operatorname{Fuk}(y)Dy=DFuk⁡(y)
whose objects are "A-type branes," supported on Lagrangian submanifolds of y y yyy, together with a choice of orientation and a flat bundle. For example, mirror symmetry should map the structure sheaf of a point in X X X\mathcal{X}X to a Lagrangian brane in y y yyy supported on a T T ∨ T^(vv)T^{\vee}T∨ fiber. The choice of a flat U ( 1 ) U ( 1 ) U(1)U(1)U(1) connection is the position of the point in the dual fiber T T TTT.
Kontsevich' homological mirror symmetry is a conjecture that the category of Bbranes on X X X\mathcal{X}X and the category of A-branes on y y yyy are equivalent,
D x D y D x ≅ D y Dx~=Dy\mathscr{D} x \cong \mathscr{D} yDx≅Dy
and that this equivalence characterizes what mirror symmetry is.

3.2. Quantum differential equation and its monodromy

Knizhnik-Zamolodchikov equation, which plays a central role in knot theory, has a geometric counterpart. In the world of mirror symmetry, there is an equally fundamental differential equation,
(3.1) i V α ( C i ) α β V β = 0 (3.1) ∂ i V α − C i α β V β = 0 {:(3.1)del_(i)V_(alpha)-(C_(i))_(alpha)^(beta)V_(beta)=0:}\begin{equation*} \partial_{i} \mathcal{V}_{\alpha}-\left(C_{i}\right)_{\alpha}^{\beta} \mathcal{V}_{\beta}=0 \tag{3.1} \end{equation*}(3.1)∂iVα−(Ci)αβVβ=0
The equation is known as the quantum differential equation of X X X\mathcal{X}X. Both the equation and its monodromy problem featured prominently, starting with the first papers on mirror symmetry, see [37] for an early account. In (3.1), ( C i ) α β = C γ i α δ η δ β C i α β = C γ i α δ η δ β (C_(i))_(alpha)^(beta)=C_(gamma_(i)alpha delta)eta^(delta beta)\left(C_{i}\right)_{\alpha}^{\beta}=C_{\gamma_{i} \alpha \delta} \eta^{\delta \beta}(Ci)αβ=Cγiαδηδβ is a connection on a vector bundle with fibers H even ( X ) = k H k ( X , k T X ) H even  ( X ) = ⨁ k   H k X , ∧ k T X ∗ H^("even ")(X)=bigoplus_(k)H^(k)(X,^^^(k)T_(X)^(**))H^{\text {even }}(\mathcal{X})=\bigoplus_{k} H^{k}\left(\mathcal{X}, \wedge^{k} T_{X}^{*}\right)Heven (X)=⨁kHk(X,∧kTX∗) over the complexified Kahler moduli space. The derivative stands for i = a i a i ∂ i = a i ∂ ∂ a i del_(i)=a_(i)(del)/(dela_(i))\partial_{i}=a_{i} \frac{\partial}{\partial a_{i}}∂i=ai∂∂ai, so that i a d = ( γ i , d ) a d ∂ i a d = γ i , d a d del_(i)a^(d)=(gamma_(i),d)a^(d)\partial_{i} a^{d}=\left(\gamma_{i}, d\right) a^{d}∂iad=(γi,d)ad for a curve of degree d H 2 ( X ) d ∈ H 2 ( X ) d inH_(2)(X)d \in H_{2}(\mathcal{X})d∈H2(X). The connection comes from quantum multiplication with classes γ i H 2 ( X ) γ i ∈ H 2 ( X ) gamma_(i)inH^(2)(X)\gamma_{i} \in H^{2}(\mathcal{X})γi∈H2(X). Given three de Rham cohomology classes on X X X\mathcal{X}X, their quantum product
(3.2) C α β γ = d 0 , d H 2 ( X ) ( α , β , γ ) d a d (3.2) C α β γ = ∑ d ≥ 0 , d ∈ H 2 ( X )   ( α , β , γ ) d a d {:(3.2)C_(alpha beta gamma)=sum_(d >= 0,d inH_(2)(X))(alpha","beta","gamma)_(d)a^(d):}\begin{equation*} C_{\alpha \beta \gamma}=\sum_{d \geq 0, d \in H_{2}(X)}(\alpha, \beta, \gamma)_{d} a^{d} \tag{3.2} \end{equation*}(3.2)Cαβγ=∑d≥0,d∈H2(X)(α,β,γ)dad
is a deformation of the classical cup product ( α , β , γ ) 0 = X α β γ ( α , β , γ ) 0 = ∫ X   α ∧ β ∧ γ (alpha,beta,gamma)_(0)=int_(X)alpha^^beta^^gamma(\alpha, \beta, \gamma)_{0}=\int_{X} \alpha \wedge \beta \wedge \gamma(α,β,γ)0=∫Xα∧β∧γ coming from Gromov-Witten theory of X : ( α , β , γ ) d X : ( α , β , γ ) d X:(alpha,beta,gamma)_(d)\mathcal{X}:(\alpha, \beta, \gamma)_{d}X:(α,β,γ)d is computed by an integral over the moduli space of degree d d ddd holomorphic maps from D = P 1 D = P 1 D=P^(1)D=\mathbb{P}^{1}D=P1 to X X X\mathcal{X}X whose image meets classes Poincaré dual of α , β α , β alpha,beta\alpha, \betaα,β and γ γ gamma\gammaγ at points. The quantum product, together with the invariant inner product η α β = η α β = eta_(alpha beta)=\eta_{\alpha \beta}=ηαβ= X α β ∫ X   α ∧ β int_(X)alpha^^beta\int_{X} \alpha \wedge \beta∫Xα∧β, gives rise to an associative algebra with structure constants C α β δ = C α β γ η γ δ C α β δ = C α β γ η γ δ C_(alpha beta)^(delta)=C_(alpha beta gamma)eta^(gamma delta)C_{\alpha \beta}{ }^{\delta}=C_{\alpha \beta \gamma} \eta^{\gamma \delta}Cαβδ=Cαβγηγδ. Flatness of the connection follows from the WDVV equations [27,31,83].
From the mirror perspective of y y yyy, the connection is the classical Gauss-Manin connection on the vector bundle over the moduli space of complex structures on y y yyy, with fibers the mid-dimensional cohomology H d ( y ) H d ( y ) H^(d)(y)H^{d}(y)Hd(y) as mirror symmetry identifies H k ( X , k T X ) H k X , ∧ k T X ∗ H^(k)(X,^^^(k)T_(X)^(**))H^{k}\left(\mathcal{X}, \wedge^{k} T_{X}^{*}\right)Hk(X,∧kTX∗) with H k ( y , d k T y ) H k y , ∧ d − k T y ∗ H^(k)(y,^^^(d-k)T_(y)^(**))H^{k}\left(y, \wedge^{d-k} T_{y}^{*}\right)Hk(y,∧d−kTy∗).

3.2.1.

Solutions to the equation live in a vector space, spanned by K-theory classes of branes [22,36,41,46]. These are B-type branes on X X X\mathcal{X}X, objects of D X D X D_(X)\mathscr{D}_{X}DX, and A-type branes on Y Y YYY, objects of D y D y Dy\mathscr{D} yDy. A characteristic feature is that the equation and its solutions mix the A- and B-type structures on the same manifold.
From the perspective of X X X\mathcal{X}X, the solutions of the quantum differential equation come from Gromov-Witten theory. They are obtained by counting holomorphic maps from a domain curve D D DDD to X X X\mathcal{X}X, where D D DDD is best thought of as an infinite cigar [ 39 , 40 ] [ 39 , 40 ] [39,40][39,40][39,40] together with insertions of a class in α H even ( X ) α ∈ H even  ∗ ( X ) alpha inH_("even ")^(**)(X)\alpha \in H_{\text {even }}^{*}(\mathcal{X})α∈Heven ∗(X) at the origin, and [ F ] K ( X ) [ F ] ∈ K ( X ) [F]in K(X)[\mathcal{F}] \in K(\mathcal{X})[F]∈K(X) at infinity. The latter is the K-theory class of a B-type brane F D x F ∈ D x FinDx\mathscr{F} \in \mathscr{D} xF∈Dx which serves as the boundary condition at the S 1 S 1 S^(1)S^{1}S1 boundary at infinity of D D DDD. In the mirror y y yyy, the A- and B-type structures get exchanged. In the interior of D D DDD, supersymmetry is preserved by B-type twist, and at the boundary at infinity we place an A-type brane L D y L ∈ D y LinDy\mathscr{L} \in \mathscr{D} yL∈Dy, whose K K KKK-theory class picks which solution of the equation we get.

3.2.2.

One of the key mirror symmetry predictions is that monodromy of the quantum differential equation gets categorified by the action of derived autoequivalences of D X D X DX\mathscr{D} XDX. It is related by mirror symmetry to the monodromy of the Gauss-Manin connection, computed by Picard-Lefshetz theory, whose categorification by D y D y Dy\mathscr{D} yDy is developed by Seidel [71].
The flat section V V V\mathcal{V}V of the connection in (3.1) has a close cousin. This is Douglas' [ 9 , 28 , 29 ] Π [ 9 , 28 , 29 ] Π [9,28,29]Pi[9,28,29] \Pi[9,28,29]Π-stability central charge function Z 0 : K ( D ) C Z 0 : K ( D ) → C Z^(0):K(D)rarrC\mathcal{Z}^{0}: K(\mathscr{D}) \rightarrow \mathbb{C}Z0:K(D)→C, whose existence motivated Bridgeland's formulation of stability conditions [17]. The Π Î  Pi\PiΠ-stability central charge Z 0 Z 0 Z^(0)Z^{0}Z0 arises from the same setting as V V V\mathcal{V}V, except one places trivial insertions at the origin of D D DDD. This implies that monodromies of V V V\mathcal{V}V and Z 0 Z 0 Z^(0)\mathcal{Z}^{0}Z0 coincide [22]. In the context of the mirror Y Y Y\mathcal{Y}Y, given any brane L D y L ∈ D y LinDy\mathscr{L} \in \mathscr{D} yL∈Dy, its central charge is simply Z 0 [ L ] = L Ω Z 0 [ L ] = ∫ L   Ω Z^(0)[L]=int_(L)Omega\mathcal{Z}^{0}[\mathscr{L}]=\int_{\mathscr{L}} \OmegaZ0[L]=∫LΩ, where Ω Î© Omega\OmegaΩ is the top holomorphic form on y y yyy. The stable objects are special Lagrangians, on which the phase of Ω Î© Omega\OmegaΩ is constant. By mirror symmetry, monodromy of Z 0 Z 0 Z^(0)\mathcal{Z}^{0}Z0 is expected to induce the action of monodromy on D x D x Dx\mathscr{D} xDx. Examples of braid group actions on the derived categories include works of Khovanov and Seidel [52], Seidel and Thomas [74], and others [18,77].
The Knizhnik-Zamolodchikov equation not only has the same flavor as the quantum differential equation, but for some very special choices of X X X\mathcal{X}X, they coincide. For the time being, we will take L g L g ^(L)g{ }^{L} \mathrm{~g}L g to be simply laced, so it coincides with its Langlands dual g g g\mathrm{g}g.

4.1. The geometry

The manifold X X X\mathcal{X}X may be described as the moduli space of G G GGG-monopoles on
(4.1) R 3 = R × C (4.1) R 3 = R × C {:(4.1)R^(3)=RxxC:}\begin{equation*} \mathbb{R}^{3}=\mathbb{R} \times \mathbb{C} \tag{4.1} \end{equation*}(4.1)R3=R×C
with prescribed singularities. The monopole group G G GGG is related to L G L G ^(L)G{ }^{L} GLG, the Chern-Simons gauge group, by Langlands or electric-magnetic duality. In Chern-Simons theory, the knots are labeled by representations of L G L G ^(L)G{ }^{L} GLG and viewed as paths of heavy particles, charged electrically under L G L G ^(L)G{ }^{L} GLG. In the geometric description, the same heavy particles appear as singular, Dirac-type monopoles of the Langlands dual group G G GGG. The fact the magnetic description is what is needed to understand categorification was anticipated by Witten [84-87].

4.1.1.

Place a singular G G GGG monopole for every finite puncture on A R × S 1 A ≅ R × S 1 A~=RxxS^(1)\mathscr{A} \cong \mathbb{R} \times S^{1}A≅R×S1, at the point on R R R\mathbb{R}R obtained by forgetting the S 1 S 1 S^(1)S^{1}S1. Singular monopole charges are elements of the cocharacter lattice of G G GGG, which Langlands duality identifies with the character lattice of L G L G ^(L)G{ }^{L} GLG. Pick the charge of the monopole to be the highest weight μ i μ i mu_(i)\mu_{i}μi of the L G L G ^(L)G{ }^{L} GLG representation V i V i V_(i)V_{i}Vi coloring the puncture. The relative positions of singular monopoles on R 3 R 3 R^(3)\mathbb{R}^{3}R3 are the moduli of the metric on X X X\mathcal{X}X, so we will hold them fixed.
The smooth monopole charge is a positive root of L G L G ^(L)G{ }^{L} GLG; choose it so that the total monopole charge is the weight ν ν nu\nuν of subspace of representation i V i ⊗ i V i ox_(i)V_(i)\otimes_{i} V_{i}⊗iVi, where the conformal blocks take values. For our current purpose, it suffices to assume
(4.2) v = i μ i a = 1 r k d a L e a (4.2) v = ∑ i   μ i − ∑ a = 1 r k   d a L e a {:(4.2)v=sum_(i)mu_(i)-sum_(a=1)^(rk)d_(a)^(L)e_(a):}\begin{equation*} v=\sum_{i} \mu_{i}-\sum_{a=1}^{r k} d_{a}^{L} e_{a} \tag{4.2} \end{equation*}(4.2)v=∑iμi−∑a=1rkdaLea
is a dominant weight; L e a L e a ^(L)e_(a){ }^{L} e_{a}Lea are the simple positive roots of L g L g ^(L)g{ }^{L} \mathrm{~g}L g. Provided μ i μ i mu_(i)\mu_{i}μi are minuscule co-weights of G G GGG and no pairs of singular monopoles coincide, the monopole moduli space X X X\mathcal{X}X is a smooth hyper-Kahler manifold of dimension
dim C ( X ) = 2 a d a dim C ⁡ ( X ) = 2 ∑ a   d a dim_(C)(X)=2sum_(a)d_(a)\operatorname{dim}_{\mathbb{C}}(\mathcal{X})=2 \sum_{a} d_{a}dimC⁡(X)=2∑ada
It is parameterized, in part, by positions of smooth monopoles on R 3 R 3 R^(3)\mathbb{R}^{3}R3.

4.1.2.

A choice of complex structure on X X X\mathcal{X}X reflects a split of R 3 R 3 R^(3)\mathbb{R}^{3}R3 as R × C R × C RxxC\mathbb{R} \times \mathbb{C}R×C. The relative positions of singular monopoles on C C C\mathbb{C}C become the complex structure moduli, and the relative positions of monopoles on R R R\mathbb{R}R the Kahler moduli.
This identifies the complexified Kahler moduli space of X X X\mathcal{X}X (where the Kahler form gets complexified by a periodic two-form) with the configuration space of n n nnn distinct punctures on A = R × S 1 A = R × S 1 A=RxxS^(1)\mathscr{A}=\mathbb{R} \times S^{1}A=R×S1, modulo overall translations, as in Figure 2 .

FIGURE 2

Punctures on A A A\mathcal{A}A correspond to singular G G GGG-monopoles on R R × C R ∈ R × C RinRxxC\mathbb{R} \in \mathbb{R} \times \mathbb{C}R∈R×C.

4.1.3.

As a hyper-Kahler manifold, X X X\mathcal{X}X has more symmetries than a typical Calabi-Yau. For its quantum cohomology to be nontrivial, and for the quantum differential equation to coincide with the K Z K Z KZ\mathrm{KZ}KZ equation, we need to work equivariantly with respect to a torus action that scales its holomorphic symplectic form
ω 2 , 0 q ω 2 , 0 ω 2 , 0 → q ω 2 , 0 omega^(2,0)rarr qomega^(2,0)\omega^{2,0} \rightarrow q \omega^{2,0}ω2,0→qω2,0
For this to be a symmetry, we will place all the singular monopoles at the origin of C ; X C ; X C;X\mathbb{C} ; \mathcal{X}C;X has a larger torus of symmetries
T = Λ × C q × T = Λ × C q × T=Lambda xxC_(q)^(xx)T=\Lambda \times \mathbb{C}_{\mathrm{q}}^{\times}T=Λ×Cq×
where Λ Î› Lambda\LambdaΛ preserves the holomorphic symplectic form, and comes from the Cartan torus of G G GGG. The equivariant parameters of the Λ Î› Lambda\LambdaΛ-action correspond to the choice of a flat L G L G ^(L)G{ }^{L} GLG connection of Chern-Simons theory on R 2 × S 1 R 2 × S 1 R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}R2×S1.

4.1.4.

The same manifold X X X\mathcal{X}X has appeared in mathematics before, as a resolution of a transversal slice in the affine Grassmannian Gr G = G ( ( z ) ) / G [ [ z ] ] Gr G = G ( ( z ) ) / G [ [ z ] ] Gr_(G)=G((z))//G[[z]]\operatorname{Gr}_{G}=G((z)) / G[[z]]GrG=G((z))/G[[z]] of G G GGG, often denoted by
(4.3) X = G r μ ν (4.3) X = G r μ → ν {:(4.3)X=Gr^( vec(mu))_(nu):}\begin{equation*} X=\mathrm{Gr}^{\vec{\mu}}{ }_{\nu} \tag{4.3} \end{equation*}(4.3)X=Grμ→ν
The two are related by thinking of monopole moduli space X X X\mathcal{X}X as obtained by a sequence of Hecke modifications of holomorphic G G GGG-bundles on C C C\mathbb{C}C [45].
Manifold X X X\mathcal{X}X is also the Coulomb branch of a 3d quiver gauge theory with N = 4 N = 4 N=4\mathcal{N}=4N=4 supersymmetry, with quiver based on the Dynkin diagram of g g ggg, see e.g. [19]. The ranks of the flavor and gauge symmetry groups are determined from the weights μ μ mu\muμ and ν ν nu\nuν.

4.1.5.

The vector μ = ( μ 1 , , μ n ) μ → = μ 1 , … , μ n vec(mu)=(mu_(1),dots,mu_(n))\vec{\mu}=\left(\mu_{1}, \ldots, \mu_{n}\right)μ→=(μ1,…,μn) in (4.3) encodes singular monopole charges, and the order in which they appear on R R R\mathbb{R}R, and ν ν nu\nuν is the total monopole charge. The ordering of entries of μ μ → vec(mu)\vec{\mu}μ→ is a choice of a chamber in the Kahler moduli. We will suppress μ μ → vec(mu)\vec{\mu}μ→ for the most part, and denote all the corresponding distinct symplectic manifolds by X X X\mathcal{X}X.

4.1.6.

By a recent theorem of Danilenko [26], the Knizhnik-Zamolodchikov equation corresponding to the Riemann surface A = R × S 1 A = R × S 1 A=RxxS^(1)\mathcal{A}=\mathbb{R} \times S^{1}A=R×S1, with punctures colored by minuscule representations V i V i V_(i)V_{i}Vi of L g L g ^(L)g{ }^{L} \mathrm{~g}L g, coincides with the quantum differential equation of the T T TTT-equivariant Gromov-Witten theory on X = G r μ ν X = G r μ → ν X=Gr^( vec(mu))_(nu)\mathcal{X}=\mathrm{Gr}^{\vec{\mu}}{ }_{\nu}X=Grμ→ν.
This has many deep consequences.

4.2. Branes and braiding

Since the K Z K Z KZ\mathrm{KZ}KZ equation is the quantum-differential equation of T T TTT-equivariant Gromov-Witten theory of X X X\mathcal{X}X, the space of its solutions gets identified with K T ( X ) K T ( X ) K_(T)(X)K_{T}(\mathcal{X})KT(X), the T T TTT-equivariant K K KKK-theory of X X X\mathcal{X}X.
This is the K K KKK-group of the category of its B-type branes, the derived category of T T TTT-equivariant coherent sheaves on X X X\mathcal{X}X,
D X = D b Coh T ( X ) D X = D b Coh T ⁡ ( X ) D_(X)=D^(b)Coh_(T)(X)\mathscr{D}_{X}=D^{b} \operatorname{Coh}_{T}(\mathcal{X})DX=DbCohT⁡(X)
This connection between the K Z K Z KZ\mathrm{KZ}KZ equation and D x D x Dx\mathscr{D} xDx is the starting point for our first solution of the categorification problem.

4.2.1.

A colored braid with n n nnn strands in A × [ 0 , 1 ] A × [ 0 , 1 ] Axx[0,1]\mathcal{A} \times[0,1]A×[0,1] has a geometric interpretation as a path in the complexified Kahler moduli of X X X\mathcal{X}X that avoids singularities, as the order of punctures on A A A\mathcal{A}A corresponds to a choice of chamber in the Kahler moduli of X X X\mathcal{X}X.
The monodromy of the quantum differential equation along this path acts on K T ( X ) K T ( X ) K_(T)(X)K_{T}(\mathcal{X})KT(X). Since the quantum differential equation coincides with the K Z K Z KZ\mathrm{KZ}KZ equation, by the theorem of [26], K T ( X ) K T ( X ) K_(T)(X)K_{T}(\mathcal{X})KT(X) becomes a module for U q ( L g ) U q L g U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(L g), corresponding to the weight ν ν nu\nuν subspace of representation V 1 V n V 1 ⊗ ⋯ ⊗ V n V_(1)ox cdots oxV_(n)V_{1} \otimes \cdots \otimes V_{n}V1⊗⋯⊗Vn
The fact that derived equivalences of D x D x Dx\mathscr{D} xDx categorify this action is not only an expectation, but also a theorem by Bezrukavnikov and Okounkov [14], whose proof makes use of quantization of X X X\mathcal{X}X in characteristic p p ppp.

4.2.2.

From physics perspective, the reason derived equivalences of D x D x Dx\mathscr{D} xDx had to categorify the action of monodromy of the quantum differential equation on K T ( X ) K T ( X ) K_(T)(X)K_{T}(\mathcal{X})KT(X) is as follows.
Braid group acts, in the σ σ sigma\sigmaσ-model on the cigar D D DDD from Section 3.2.1, by letting the moduli of X X X\mathcal{X}X vary according to the braid near the boundary at infinity. The Euclidean time, running along the cigar, is identified with the time along the braid. This leads to a Berry phase-type problem studied by Cecotti and Vafa [22]. It follows that the σ σ sigma\sigmaσ-model on the annulus, with moduli that vary according to the braid, computes the matrix element of the monodromy B B B\mathfrak{B}B, picked out by a pair of branes F 0 F 0 F_(0)\mathscr{F}_{0}F0 and F 1 F 1 F_(1)\mathscr{F}_{1}F1 at its boundaries.
The σ σ sigma\sigmaσ-model on the same Euclidian annulus, where we take the time to run around S 1 S 1 S^(1)S^{1}S1 instead, computes the index of the supercharge Q Q QQQ preserved by the two branes. The cohomology of Q Q QQQ is computed by D x D x Dx\mathscr{D} xDx as its most basic ingredient, the space of morphisms
Hom D X , ( B F 0 , F 1 ) Hom D X ∗ , ∗ ⁡ B F 0 , F 1 Hom_(DX)^(**,**)(BF_(0),F_(1))\operatorname{Hom}_{\mathscr{D} \mathcal{X}}^{*, *}\left(\mathscr{B} \mathcal{F}_{0}, \mathscr{F}_{1}\right)HomDX∗,∗⁡(BF0,F1)
between the branes. This is the space of supersymmetric ground states of the σ σ sigma\sigmaσ-model on a strip, obtained by cutting the annulus open. We took here all the variations of moduli to happen near one boundary, at the expense of changing a boundary condition from F 0 F 0 F_(0)\mathscr{F}_{0}F0 to B F 0 B F 0 BF_(0)\mathscr{B} \mathcal{F}_{0}BF0. This does not affect the homology [ 1 , 35 ] [ 1 , 35 ] [1,35][1,35][1,35], for the very same reason the theory depends on the homotopy type of the braid only. Per construction, the graded Euler characteristic of the homology theory, computed by closing the strip back up to the annulus, is the braiding matrix element,
(4.4) χ ( B F 0 , F 1 ) = ( B V 0 , V 1 ) (4.4) χ B F 0 , F 1 = B V 0 , V 1 {:(4.4)chi(BF_(0),F_(1))=(BV_(0),V_(1)):}\begin{equation*} \chi\left(\mathscr{B} \mathcal{F}_{0}, \mathscr{F}_{1}\right)=\left(\mathscr{B} \mathcal{V}_{0}, \mathcal{V}_{1}\right) \tag{4.4} \end{equation*}(4.4)χ(BF0,F1)=(BV0,V1)
between the conformal blocks V 0 , 1 = V [ F 0 , 1 ] V 0 , 1 = V F 0 , 1 V_(0,1)=V[F_(0,1)]\mathcal{V}_{0,1}=\mathcal{V}\left[\mathcal{F}_{0,1}\right]V0,1=V[F0,1] of the two branes.
Thus, by viewing the same Euclidian annulus in two different ways, we learn that the braid group action on the derived category
(4.5) B : D x μ D x μ (4.5) B : D x μ → → D x μ → ′ {:(4.5)B:Dx_( vec(mu))rarrDx_( vec(mu)^(')):}\begin{equation*} \mathscr{B}: \mathscr{D} x_{\vec{\mu}} \rightarrow \mathscr{D} x_{\vec{\mu}^{\prime}} \tag{4.5} \end{equation*}(4.5)B:Dxμ→→Dxμ→′
manifestly categorifies the monodromy matrix B U q ( L g ) B ∈ U q L g BinU_(q)(^(L)(g))\mathfrak{B} \in U_{\mathfrak{q}}\left({ }^{L} \mathrm{~g}\right)B∈Uq(L g) of the K Z K Z KZ\mathrm{KZ}KZ equation.
The quantum U q ( L g ) U q L g U_(q)(^(L)(g))U_{\mathcal{q}}\left({ }^{L} \mathrm{~g}\right)Uq(L g) invariants of knots and links are matrix elements of the braiding matrix B B B\mathscr{B}B, so they too will be categorified by D x D x Dx\mathscr{D} xDx, provided we can identify objects U D x U ∈ D x U inDxU \in \mathscr{D} xU∈Dx which serve as cups and caps.
Conformal blocks corresponding to cups and caps are defined using fusion [62]. The geometric analogue of fusion, in terms of X X X\mathcal{X}X and its category of branes, was shown in [1] to be the existence of certain perverse filtrations on D x D x Dx\mathscr{D} xDx, defined by abstractly by Chuang and Rouqiuer [24]. The utility of perverse filtrations for understanding the action of braiding on D x D x Dx\mathscr{D} xDx parallels the utility of fusion in describing the action of braiding in conformal field theory. In particular, it leads to identification of the cup and cap branes U U U\boldsymbol{U}U we need, and a

4.3.1.

As we bring a pair of punctures at a i a i a_(i)a_{i}ai and a j a j a_(j)a_{j}aj on A A A\mathcal{A}A together, we get a new natural basis of solutions to the K Z K Z KZ\mathrm{KZ}KZ equation, called the fusion basis, whose virtue is that it diagonalizes braiding. The possible eigenvectors are labeled by the representations
(4.6) V i V j = m = 0 m max V k m (4.6) V i ⊗ V j = ⨂ m = 0 m max   V k m {:(4.6)V_(i)oxV_(j)=⨂_(m=0)^(m_(max))V_(k_(m)):}\begin{equation*} V_{i} \otimes V_{j}=\bigotimes_{m=0}^{m_{\max }} V_{k_{m}} \tag{4.6} \end{equation*}(4.6)Vi⊗Vj=⨂m=0mmaxVkm
that occur in the tensor product of representations V i V i V_(i)V_{i}Vi and V j V j V_(j)V_{j}Vj labeling the punctures. Because V i V i V_(i)V_{i}Vi and V j V j V_(j)V_{j}Vj are minuscule representations, the nonzero multiplicities on the right-hand side are all equal to 1 . The cap arises as a special case, obtained by starting with a pair of conjugate representations V i V i V_(i)V_{i}Vi and V i V i ⋆ V_(i)^(***)V_{i}^{\star}Vi⋆, and picking the trivial representation in their tensor product.

4.3.2.

From perspective of X X X\mathcal{X}X, a pair of singular monopoles of charges μ i μ i mu_(i)\mu_{i}μi and μ j μ j mu_(j)\mu_{j}μj are coming together on R R R\mathbb{R}R, as in Figure 2, and we approach a wall in Kahler moduli at which X X X\mathcal{X}X develops a singularity. At the singularity, a collection of cycles vanishes. This is due to monopole bubbling phenomena described by Kapustin and Witten in [45].
The types of monopole bubbling that can occur are labeled by representations V k m V k m V_(k_(m))V_{k_{m}}Vkm that occur in the tensor product V i V j V i ⊗ V j V_(i)oxV_(j)V_{i} \otimes V_{j}Vi⊗Vj. The moduli space of monopoles whose positions we need to tune for the bubbling of type V k m V k m V_(k_(m))V_{k_{m}}Vkm to occur is Gr μ k m ( μ i , μ j ) = T F k m Gr μ k m μ i , μ j = T ∗ F k m Gr_(mu_(k_(m)))^((mu_(i),mu_(j)))=T^(**)F_(k_(m))\operatorname{Gr}_{\mu_{k_{m}}}^{\left(\mu_{i}, \mu_{j}\right)}=T^{*} F_{k_{m}}Grμkm(μi,μj)=T∗Fkm, where μ k m μ k m mu_(k_(m))\mu_{k_{m}}μkm is the highest weight of V k m V k m V_(k_(m))V_{k_{m}}Vkm. This space is transverse to the locus where exactly μ i + μ j μ k m μ i + μ j − μ k m mu_(i)+mu_(j)-mu_(k_(m))\mu_{i}+\mu_{j}-\mu_{k_{m}}μi+μj−μkm monopoles have bubbled off [1]. It has a vanishing cycle F k m F k m F_(k_(m))F_{k_{m}}Fkm, corresponding to the representation V k m V k m V_(k_(m))V_{k_{m}}Vkm, as its zero section. (Viewing X X X\mathcal{X}X as the Coulomb branch, monopole bubbling is related to partial Higgsing phenomena.)

4.3.3.

Conformal blocks which diagonalize the action of braiding do not in general come from actual objects of the derived category D x D x D_(x)\mathscr{D}_{x}Dx. As is well known from Picard-Lefshetz theory, eigensheaves of braiding, branes on which the braiding acts only by degree shifts B E = E [ D E ] { C E } B E = E D E C E BE=E[D_(E)]{C_(E)}\mathscr{B} \mathcal{E}=\mathcal{E}\left[D_{\mathcal{E}}\right]\left\{C_{\mathcal{E}}\right\}BE=E[DE]{CE}, are very rare.
What one gets instead [1] is a filtration
(4.7) D k 0 D k 1 D k max = D x (4.7) D k 0 ⊂ D k 1 ⊂ ⋯ ⊂ D k max = D x {:(4.7)D_(k_(0))subD_(k_(1))sub cdots subD_(k_(max))=Dx:}\begin{equation*} \mathscr{D}_{k_{0}} \subset \mathscr{D}_{k_{1}} \subset \cdots \subset \mathscr{D}_{k_{\max }}=\mathscr{D} x \tag{4.7} \end{equation*}(4.7)Dk0⊂Dk1⊂⋯⊂Dkmax=Dx
by the order of vanishing of the Π Î  Pi\PiΠ-stability central charge Z 0 : K ( X ) C Z 0 : K ( X ) → C Z^(0):K(X)rarrC\mathcal{Z}^{0}: K(\mathcal{X}) \rightarrow \mathbb{C}Z0:K(X)→C. More precisely, one gets a pair of such filtrations, one on each side of the wall. Crossing the wall preserves the filtration, but has the effect of mixing up branes at a given order in the filtration, with those at lower orders, whose central charge vanishes faster. Because X X X\mathcal{X}X is hyper-Kahler, the Π Î  Pi\PiΠ-stability central charge is given in terms of classical geometry (by Eq. (4.7) of [1]).
The existence of the filtration with the stated properties follows from the existence of the equivariant central charge function Z Z Z\mathcal{Z}Z,
(4.8) Z : K T ( X ) C (4.8) Z : K T ( X ) → C {:(4.8)Z:K_(T)(X)rarrC:}\begin{equation*} \mathcal{Z}: K_{T}(\mathcal{X}) \rightarrow \mathbb{C} \tag{4.8} \end{equation*}(4.8)Z:KT(X)→C
and the fact the action of braiding on K T ( X ) K T ( X ) K_(T)(X)K_{T}(\mathcal{X})KT(X) lifts to the action on D X D X DX\mathscr{D} XDX, by the theorem of [14]. The equivariant central charge Z Z Z\mathcal{Z}Z is computed by the equivariant Gromov-Witten theory on X X X\mathcal{X}X in a manner analogous to V V V\mathcal{V}V, starting with the σ σ sigma\sigmaσ-model on the cigar D D DDD except with no insertion at its tip. It reduces to the Π Î  Pi\PiΠ-stability central charge Z 0 Z 0 Z^(0)\mathcal{Z}^{0}Z0 by turning the equivariant parameters off.

4.3.4.

While B B B\mathscr{B}B has few eigensheaves in D x D x Dx\mathscr{D} \mathcal{x}Dx, it acts by degree shifts
(4.9) B : D k m / D k m 1 D k m / D k m 1 [ D k m ] { C k m } (4.9) B : D k m / D k m − 1 → D k m / D k m − 1 D k m C k m {:(4.9)B:D_(k_(m))//D_(k_(m-1))rarrD_(k_(m))//D_(k_(m-1))[D_(k_(m))]{C_(k_(m))}:}\begin{equation*} \mathscr{B}: \mathscr{D}_{k_{m}} / \mathscr{D}_{k_{m-1}} \rightarrow \mathscr{D}_{k_{m}} / \mathscr{D}_{k_{m-1}}\left[D_{k_{m}}\right]\left\{C_{k_{m}}\right\} \tag{4.9} \end{equation*}(4.9)B:Dkm/Dkm−1→Dkm/Dkm−1[Dkm]{Ckm}
on the quotient subcategories. The degree shifts may be read off from the eigenvectors of the action of braiding on the equivariant central charge function Z Z Z\mathcal{Z}Z. As the punctures at a i a i a_(i)a_{i}ai and a j a j a_(j)a_{j}aj come together, the eigenvector corresponding to the representation V k m V k m V_(k_(m))V_{k_{m}}Vkm in (4.6), vanishes as [1]
Z k m = ( a i a j ) D k m + C k m / κ × finite Z k m = a i − a j D k m + C k m / κ ×  finite  Z_(k_(m))=(a_(i)-a_(j))^(D_(k_(m))+C_(k_(m))//kappa)xx" finite "\mathcal{Z}_{k_{m}}=\left(a_{i}-a_{j}\right)^{D_{k_{m}}+C_{k_{m}} / \kappa} \times \text { finite }Zkm=(ai−aj)Dkm+Ckm/κ× finite 
It follows that braiding a i a i a_(i)a_{i}ai and a j a j a_(j)a_{j}aj counterclockwise acts by
Z k m ( 1 ) D k m q 1 2 C k m Z k m Z k m → ( − 1 ) D k m q 1 2 C k m Z k m Z_(k_(m))rarr(-1)^(D_(km))q^((1)/(2)C_(k_(m)))Z_(k_(m))\mathcal{Z}_{k_{m}} \rightarrow(-1)^{D_{k m}} \mathfrak{q}^{\frac{1}{2} C_{k_{m}}} \mathcal{Z}_{k_{m}}Zkm→(−1)Dkmq12CkmZkm
The cohomological degree shift D k m = dim C F k m D k m = dim C ⁡ F k m D_(k_(m))=dim_(C)F_(k_(m))D_{k_{m}}=\operatorname{dim}_{\mathbb{C}} F_{k_{m}}Dkm=dimC⁡Fkm is by the dimension of the vanishing cycle. The equivariant degree shift C k m C k m C_(k_(m))C_{k_{m}}Ckm is essentially the one familiar from the action of braiding on conformal blocks of L g ^ L g ^ widehat(L_(g))\widehat{L_{\mathrm{g}}}Lg^ in the fusion basis [1].

4.3.5.

The derived equivalences of this type are the perverse equivalences of Chuang and Rouquier [23, 24]. They envisioned them as a way to describe derived equivalences which come from variations of Bridgeland stability conditions, but with few examples from geometry.
Traditionally, braid group actions on derived categories of coherent sheaves, or Bbranes, are fairly difficult to describe, see for example [20,21]. Braid group actions on the categories of A-branes are much easier to understand, via Picard-Lefshetz theory and its categorical uplifts [71], see e.g. [52,77]. The theory of variations of stability conditions, by Douglas and Bridgeland, was invented to bridge the two [ 9 , 29 ] [ 9 , 29 ] [9,29][9,29][9,29].

4.3.6.

As a by-product, we learn that conformal blocks describing collections cups or caps colored by minuscule representations, come from branes in D x D x Dx\mathscr{D} xDx which have a simple geometric meaning [ 1 ] [ 1 ] [1][1][1].
Take X = Gr 0 ( μ 1 , μ 1 , , μ d , μ d ) X = Gr 0 μ 1 , μ 1 ∗ , … , μ d , μ d ∗ X=Gr_(0)^((mu_(1),mu_(1)^(**),dots,mu_(d),mu_(d)^(**)))\mathcal{X}=\operatorname{Gr}_{0}^{\left(\mu_{1}, \mu_{1}^{*}, \ldots, \mu_{d}, \mu_{d}^{*}\right)}X=Gr0(μ1,μ1∗,…,μd,μd∗) corresponding to A A A\mathcal{A}A with n = 2 d n = 2 d n=2dn=2 dn=2d punctures, colored by pairs of complex conjugate, minuscule representations V i V i V_(i)V_{i}Vi and V i V i ∗ V_(i)^(**)V_{i}^{*}Vi∗. We get a vanishing cycle U U UUU in X X X\mathcal{X}X which is a product of d d ddd minuscule Grassmannians,
U = G / P 1 × × G / P d U = G / P 1 × ⋯ × G / P d U=G//P_(1)xx cdots xx G//P_(d)U=G / P_{1} \times \cdots \times G / P_{d}U=G/P1×⋯×G/Pd
where P i P i P_(i)P_{i}Pi is the maximal parabolic subgroup of G G GGG associated to representation V i V i V_(i)V_{i}Vi. This vanishing cycle embeds in X X X\mathcal{X}X as a compact holomorphic Lagrangian, so in the neighborhood of U U UUU, we can model X X X\mathcal{X}X as T U T ∗ U T^(**)UT^{*} UT∗U. The structure sheaf
U = O U D X U = O U ∈ D X U=O_(U)inD_(X)U=\mathcal{O}_{U} \in \mathscr{D}_{X}U=OU∈DX
of U U UUU is the brane we are after. The Grassmannian G / P i G / P i G//P_(i)G / P_{i}G/Pi is the cycle that vanishes when a single pair of singular monopoles of charges μ i μ i mu_(i)\mu_{i}μi and μ i μ i ∗ mu_(i)^(**)\mu_{i}^{*}μi∗ come together, as Gr 0 ( μ i , μ i ) = Gr 0 μ i , μ i ∗ = Gr_(0)^((mu_(i),mu_(i)^(**)))=\operatorname{Gr}_{0}^{\left(\mu_{i}, \mu_{i}^{*}\right)}=Gr0(μi,μi∗)= T G / P i T ∗ G / P i T^(**)G//P_(i)T^{*} G / P_{i}T∗G/Pi.
The brane U U U\mathcal{U}U lives at the very bottom of a d d ddd-fold filtration which D x D x Dx\mathscr{D} xDx develops at the intersection of d d ddd walls in the Kahler moduli of X X X\mathcal{X}X corresponding to bringing punctures together pairwise. It follows U U U\mathcal{U}U is the eigensheaf of braiding each pair of matched endpoints. It is extremely special, for the same reason the trivial representation is special.

4.3.7.

Just as fusion provides the right language to understand the action of braiding in conformal field theory, the perverse filtrations provide the right language to describe the action of braiding on derived categories. Using perverse filtrations and the very special properties of the vanishing cycle branes U D x U ∈ D x U inDxU \in \mathscr{D} xU∈Dx, one gets the following theorem [1]:
Theorem 1. For any simply laced Lie algebra L g L g ^(L)g{ }^{L} \mathfrak{g}Lg, the homology groups
Hom D , ( B U , U ) Hom D ∗ , ∗ ⁡ ( B U , U ) Hom_(D)^(**,**)(BU,U)\operatorname{Hom}_{\mathscr{D}}^{*, *}(\mathscr{B} U, U)HomD∗,∗⁡(BU,U)
categorify U q ( L g ) U q L g U_(q)(^(L)g)U_{\mathfrak{q}}\left({ }^{L} \mathfrak{g}\right)Uq(Lg) quantum link invariants, and are themselves link invariants.

4.3.8.

As an illustration, proving that (the equivalent of) the pitchfork move in the figure below holds in D x D x Dx\mathscr{D} xDx

FIGURE 3

A move equivalent to the pitchfork move.
requires showing that we have a derived equivalence
(4.10) B C i C i (4.10) B ∘ C i ≅ C i ′ ′ {:(4.10)B@C_(i)~=C_(i)^(''):}\begin{equation*} \mathscr{B} \circ \mathscr{C}_{i} \cong \mathscr{C}_{i}^{\prime \prime} \tag{4.10} \end{equation*}(4.10)B∘Ci≅Ci′′
where C i C i C_(i)\mathscr{C}_{i}Ci and C i C i ′ ′ C_(i)^('')\mathscr{C}_{i}^{\prime \prime}Ci′′ are cup functors on the right and the left in Figure 3, respectively. They increase the number of strands by two and map
C i : D X n 2 D X n and C i : D X n 2 D X n C i : D X n − 2 → D X n  and  C i ′ ′ : D X n − 2 → D X n ′ ′ C_(i):D_(X_(n-2))rarrD_(X_(n))quad" and "quadC_(i)^(''):D_(X_(n-2))rarrD_(X_(n)^(''))\mathscr{C}_{i}: \mathscr{D}_{X_{n-2}} \rightarrow \mathscr{D}_{X_{n}} \quad \text { and } \quad \mathscr{C}_{i}^{\prime \prime}: \mathscr{D}_{X_{n-2}} \rightarrow \mathscr{D}_{X_{n}^{\prime \prime}}Ci:DXn−2→DXn and Ci′′:DXn−2→DXn′′
where the subscript serves to indicate the number of strands. The functor B B B\mathscr{B}B is the equivalence of categories from the theorem of [14]
B : D X n D X n B : D X n → D X n ′ ′ B:D_(X_(n))rarrD_(X_(n)^(''))\mathscr{B}: \mathscr{D}_{X_{n}} \rightarrow \mathscr{D}_{X_{n}^{\prime \prime}}B:DXn→DXn′′
corresponding to braiding V k ( a k ) V k a k V_(k)(a_(k))V_{k}\left(a_{k}\right)Vk(ak) with V i ( a i ) V i ( a j ) V i a i ⊗ V i ∗ a j V_(i)(a_(i))oxV_(i)^(**)(a_(j))V_{i}\left(a_{i}\right) \otimes V_{i}^{*}\left(a_{j}\right)Vi(ai)⊗Vi∗(aj) where V i V i V_(i)V_{i}Vi and V i V i ∗ V_(i)^(**)V_{i}^{*}Vi∗ color the red and V k V k V_(k)V_{k}Vk the black strand in Figure 3.
To prove the identity (4.10) note that
(4.11) C i D x n 2 D x n and C i D x n 2 D X n (4.11) C i D x n − 2 ⊂ D x n  and  C i ′ ′ D x n − 2 ⊂ D X n ′ ′ {:(4.11)C_(i)Dx_(n-2)subDx_(n)quad" and "quadC_(i)^('')D_(x_(n-2))subD_(X_(n)^('')):}\begin{equation*} \mathscr{C}_{i} \mathscr{D} x_{n-2} \subset \mathscr{D} x_{n} \quad \text { and } \quad \mathscr{C}_{i}^{\prime \prime} \mathscr{D}_{x_{n-2}} \subset \mathscr{D}_{X_{n}^{\prime \prime}} \tag{4.11} \end{equation*}(4.11)CiDxn−2⊂Dxn and Ci′′Dxn−2⊂DXn′′
are the subcategories which are the bottom-most part of the double filtrations of D x n D x n Dx_(n)\mathscr{D} x_{n}Dxn and D X n D X n ′ ′ D_(X_(n)^(''))\mathscr{D}_{X_{n}^{\prime \prime}}DXn′′, corresponding to the intersection of walls at which the three punctures come together. By the definition of perverse filtrations, the functor B B B\mathscr{B}B acts at a bottom part of a double filtration at most by degree shifts. The degree shifts are trivial too, since if they were not, the relation we are after would not hold even in conformal field theory, and we know it does. To complete the proof, one recalls that a perverse equivalence that acts by degree shifts that are trivial is an equivalence of categories [24].
Proofs of invariance under the Reidermeister 0 and the framed Reidermeister I moves are similar. The invariance under Reidermeister II and III moves follows from the theorem of [14]. One should compare this to proofs of the same relations in [20, 21], which are more technical and less general.

4.3.9.

An elementary consequence is a geometric explanation of mirror symmetry which relates the U q ( L g ) U q L g U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(L g) invariants of a link K K KKK and its mirror reflection K K ∗ K^(**)K^{*}K∗.
It is a consequence of a basic property of D X D X D_(X)\mathscr{D}_{X}DX, Serre duality. Serre duality implies the isomorphism of homology groups on X X X\mathcal{X}X which is a 2 d 2 d 2d2 d2d complex-dimensional Calabi-Yau manifold,
(4.12) Hom D X ( B U , U [ M ] { J 0 , J } ) = Hom D x ( B U , U [ 2 d M ] { d J 0 , J } ) (4.12) Hom D X ⁡ B U , U [ M ] J 0 , J → = Hom D x ⁡ B U , U [ 2 d − M ] − d − J 0 , − J → {:(4.12)Hom_(DX)(BU,U[M]{J_(0),( vec(J))})=Hom_(Dx)(BU,U[2d-M]{-d-J_(0),-( vec(J))}):}\begin{equation*} \operatorname{Hom}_{\mathscr{D} X}\left(\mathscr{B} U, U[M]\left\{J_{0}, \vec{J}\right\}\right)=\operatorname{Hom}_{\mathscr{D} x}\left(\mathscr{B} U, U[2 d-M]\left\{-d-J_{0},-\vec{J}\right\}\right) \tag{4.12} \end{equation*}(4.12)HomDX⁡(BU,U[M]{J0,J→})=HomDx⁡(BU,U[2d−M]{−d−J0,−J→})
The equivariant degree shift comes from the fact the unique holomorphic section of the canonical bundle has weight d d ddd under the C q × T C q × ⊂ T C_(q)^(xx)subT\mathbb{C}_{\mathrm{q}}^{\times} \subset \mathrm{T}Cq×⊂T action. Mirror symmetry follows by taking Euler characteristic of both sides [1].

4.4. Algebra from B-branes

Bezrukavnikov and Kaledin, using quantization in characteristic p p ppp, constructed a tilting vector bundle T T T\mathcal{T}T, on any X X X\mathcal{X}X which is a symplectic resolution [ 12 , 13 , 43 , 44 ] [ 12 , 13 , 43 , 44 ] [12,13,43,44][12,13,43,44][12,13,43,44]. Its endomorphism algebra
A = Hom D X ( T , T ) A = Hom D X ∗ ⁡ ( T , T ) A=Hom_(DX)^(**)(T,T)\mathscr{A}=\operatorname{Hom}_{\mathscr{D} X}^{*}(\mathcal{T}, \mathcal{T})A=HomDX∗⁡(T,T)
is an ordinary associative algebra, graded only by equivariant degrees. The derived category D A D A D_(A)\mathscr{D}_{\mathscr{A}}DA of its modules is equivalent to D X D X D_(X)\mathscr{D}_{X}DX,
D X D A D X ≅ D A D_(X)~=D_(A)\mathscr{D}_{X} \cong \mathscr{D}_{\mathscr{A}}DX≅DA
essentially per definition.
Webster recently computed the algebra A A A\mathscr{A}A for our X X X\mathcal{X}X [80], and showed that it coincides with a cylindrical version of the KLRW algebra from [78]. Working with the cylindrical KLRW algebra, as opposed to the ordinary one, leads to invariants of links in R 2 × S 1 R 2 × S 1 R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}R2×S1 and not just in R 3 R 3 R^(3)\mathbb{R}^{3}R3. The KLRW algebra generalizes the algebras of Khovanov and Lauda [50] and Rouquier [68]. The cylindrical version of the KLR algebra corresponds to X X X\mathcal{X}X which is a Coulomb branch of a pure 3D gauge theory.

4.4.1.

The description of link homologies via D X = Coh T ( X ) D X = Coh T ⁡ ( X ) D_(X)=Coh_(T)(X)\mathscr{D}_{X}=\operatorname{Coh}_{T}(\mathcal{X})DX=CohT⁡(X) provides a geometric meaning of homological U q ( L g ) U q L g U_(q)(^(L)g)U_{\mathfrak{q}}\left({ }^{L} \mathfrak{g}\right)Uq(Lg) link invariants. Even so, without further input, the description of link homologies either in terms of D X D X D_(X)\mathscr{D}_{X}DX or D A D A D_(A)\mathscr{D}_{\mathscr{A}}DA is purely formal. With the help of (equivariant) homological mirror symmetry, we will give a description of link homology groups which is explicit and explicitly computable; in this sense, link homology groups come to life in the mirror.

5. MIRROR SYMMETRY FOR MONOPOLE MODULI SPACE

In the very best instances, homological mirror symmetry relating D y D y Dy\mathscr{D} yDy and D x D x Dx\mathscr{D} xDx can be made manifest, by showing that each is equivalent to D A D A D_(A)\mathscr{D}_{\mathscr{A}}DA, the derived category of modules
of the same associative algebra A A A\mathscr{A}A,
(5.1) D x D A D y (5.1) D x ≅ D A ≅ D y {:(5.1)D_(x)~=D_(A)~=Dy:}\begin{equation*} \mathscr{D}_{x} \cong \mathscr{D}_{\mathscr{A}} \cong \mathscr{D} y \tag{5.1} \end{equation*}(5.1)Dx≅DA≅Dy
The algebra
A = Hom D ( T , T ) A = Hom D ∗ ⁡ ( T , T ) A=Hom_(D)^(**)(T,T)\mathscr{A}=\operatorname{Hom}_{\mathscr{D}}^{*}(\mathcal{T}, \mathcal{T})A=HomD∗⁡(T,T)
is the endomorphism algebra of a set of branes T = C T e T = ⨁ C   T e T=bigoplus_(C)Te\mathcal{T}=\bigoplus_{\mathscr{C}} \mathcal{T} \mathscr{e}T=⨁CTe, which generate D x D x Dx\mathscr{D} xDx and D y D y Dy\mathscr{D} yDy. For economy, we will be denoting branes related by mirror symmetry by the same letter.
An elementary example [10] is mirror symmetry relating a pair of infinite cylinders, X = C × X = C × X=C^(xx)\mathcal{X}=\mathbb{C}^{\times}X=C×and y = R × S 1 y = R × S 1 y=RxxS^(1)y=\mathbb{R} \times S^{1}y=R×S1, whose torus fibers are dual S 1 S 1 S^(1)S^{1}S1, s. Both D x D x Dx\mathscr{D} xDx, the derived category of coherent sheaves on X X X\mathcal{X}X, and D y D y Dy\mathscr{D} yDy, based on the wrapped Fukaya category, are generated by a single object T T T\mathcal{T}T, a flat line bundle on X X X\mathcal{X}X and a real-line Lagrangian on Y Y Y\mathscr{Y}Y. Their algebras of open strings are the same, equal to the algebra A = C [ x ± 1 ] A = C x ± 1 A=C[x^(+-1)]\mathscr{A}=\mathbb{C}\left[x^{ \pm 1}\right]A=C[x±1] of holomorphic functions on the cylinder.
X X X\mathcal{X}X
FIGURE 4
A simple example of manifest mirror symmetry.

5.1. The algebra for homological mirror symmetry

In our setting, the generator T T T\mathcal{T}T of D x D x Dx\mathscr{D} xDx is the tilting generator of Bezrukavnikov and Kaledin from Section 4.4. Webster's proof of the equivalence of categorification of U q ( L g ) U q L g U_(q)(^(L)g)U_{\mathfrak{q}}\left({ }^{L} \mathfrak{g}\right)Uq(Lg) link invariants and B-type branes on X X X\mathcal{X}X and via the cKLRW algebra A A A\mathscr{A}A should be understood as the first of the two equivalences in (5.1).

5.1.1.

The mirror y y yyy of X X X\mathcal{X}X is the moduli space of G G GGG monopoles, of the same charges as X X XXX except on R 2 × S 1 R 2 × S 1 R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}R2×S1 instead of on R 3 R 3 R^(3)\mathbb{R}^{3}R3, with only complex and no Kahler moduli turned on, and equipped with a potential [2]. Without the potential, the mirror to y y yyy would be another moduli space of G G GGG monopoles on R 2 × S 1 R 2 × S 1 R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}R2×S1.
The theory based on D y D y Dy\mathscr{D} yDy, the derived Fukaya-Seidel category of y y yyy, is in the same spirit as the work of Seidel and Smith [72]. They pioneered geometric approaches to link homology, but produced a only singly graded theory, known as symplectic Khovanov homology. The computation of D y D y Dy\mathscr{D} yDy, which makes mirror symmetry in (5.1) manifest, is given in the joint work with Danilenko, Li, and Zhou [4].

5.2. The core of the monopole moduli space

Working equivariantly with respect to a C q × C q × C_(q)^(xx)\mathbb{C}_{\mathrm{q}}^{\times}Cq×-symmetry which scales the holomorphic symplectic form of X X X\mathcal{X}X, all the information about its geometry should be encoded in a core locus preserved by such actions.
The core X X XXX is a singular holomorphic Lagrangian in X X X\mathcal{X}X which is the union of supports of all stable envelopes [7,61]. Equivalently, X X XXX is the union of all attracting sets of Λ Î› Lambda\LambdaΛ-torus actions on X X X\mathcal{X}X, where we let Λ Î› Lambda\LambdaΛ vary over all chambers. If we view X X X\mathcal{X}X as the monopole moduli space, we can put this more simply: X X XXX is the locus where all the monopoles, singular or not, are at the origin of C C C\mathbb{C}C in R × C R × C RxxC\mathbb{R} \times \mathbb{C}R×C. Viewing it as a Coulomb branch, X X XXX is the locus at which the complex scalar fields in vector multiplets vanish.
We will define the equivariant mirror Y Y YYY of X X X\mathcal{X}X to be the ordinary mirror of its core, so we have
Working equivariantly with respect to the T T TTT-action on X X X\mathcal{X}X, the equivariant mirror gets a potential W W WWW, making the theory on Y Y YYY into a Landau-Ginsburg model. While X X XXX embeds into X X X\mathcal{X}X as a holomorphic Lagrangian of dimension d , y d , y d,yd, yd,y fibers over Y Y YYY with holomorphic Lagrangian ( C × ) d C × d (C^(xx))^(d)\left(\mathbb{C}^{\times}\right)^{d}(C×)d fibers.

5.2.1.

A model example is X X X\mathcal{X}X which is the resolution of an A n 1 A n − 1 A_(n-1)A_{n-1}An−1 hypersurface singularity, u v = z n ; X u v = z n ; X uv=z^(n);Xu v=z^{n} ; \mathcal{X}uv=zn;X is the moduli space of a single smooth G = S U ( 2 ) / Z 2 G = S U ( 2 ) / Z 2 G=SU(2)//Z_(2)G=\mathrm{SU}(2) / \mathbb{Z}_{2}G=SU(2)/Z2 monopole, in the presence of n n nnn singular ones. The core X X XXX is a collection of n 1 P 1 n − 1 P 1 n-1P^(1)n-1 \mathbb{P}^{1}n−1P1 's with a pair of infinite discs attached, as in Figure 5.
FIGURE 5
Core X X XXX of a resolution of the A n 1 A n − 1 A_(n-1)A_{n-1}An−1 singularity.
The ordinary mirror y y yyy of X X X\mathcal{X}X is the complex structure deformation of the "multiplicative" A n 1 A n − 1 A_(n-1)A_{n-1}An−1 surface singularity, with a potential which we will not need. y y yyy is a C × C × C^(xx)\mathbb{C}^{\times}C×fibration over Y Y YYY which is itself an infinite cylinder, a copy of C × C × C^(xx)\mathbb{C}^{\times}C×with n n nnn points deleted. At the marked points, the C × C × C^(xx)\mathbb{C}^{\times}C×fibers degenerate. There are n 1 n − 1 n-1n-1n−1 Lagrangian spheres in y y yyy, which are mirror to n 1 P 1 n − 1 P 1 n-1P^(1)n-1 \mathbb{P}^{1}n−1P1 's in X X XXX. They project to Lagrangians in Y Y YYY which begin and end at the punctures.

5.2.2.

The model example corresponds to L G = S U ( 2 ) L G = S U ( 2 ) ^(L)G=SU(2){ }^{L} G=\mathrm{SU}(2)LG=SU(2) Chern-Simons theory on R 2 × S 1 R 2 × S 1 R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}R2×S1, and N 1 2 ^ N 1 2 ^ widehat(N1_(2))\widehat{\mathfrak{N 1}_{2}}N12^ conformal blocks on A = R × S 1 A = R × S 1 A=RxxS^(1)\mathcal{A}=\mathbb{R} \times S^{1}A=R×S1. The n n nnn punctures on A A A\mathcal{A}A are colored by the fundamental, two-dimensional representation V 1 / 2 V 1 / 2 V_(1//2)V_{1 / 2}V1/2 of N u 2 N u 2 Nu_(2)\mathfrak{N u}_{2}Nu2, and we take the subspace of weight
FIGURE 6
Lagrangian spheres in y y y\boldsymbol{y}y mirror the vanishing P 1 P 1 P^(1)\mathbb{P}^{1}P1 's in X X X\mathcal{X}X.
one level below the highest. Note that Y Y YYY coincides with the Riemann surface A A A\mathcal{A}A where the conformal blocks live. This is not an accident.
In the model example, both X X XXX and Y Y YYY are S 1 S 1 S^(1)S^{1}S1 fibrations over R R R\mathbb{R}R with n n nnn marked points. At the marked points, the S 1 S 1 S^(1)S^{1}S1 fibers of X X XXX degenerate. In Y Y YYY, this is mirrored by fibers that decompactify, due to points which are deleted.

5.2.3.

More generally, for X = G r μ ν X = G r μ → ν X=Gr^( vec(mu))_(nu)\mathcal{X}=\mathrm{Gr}^{\vec{\mu}}{ }_{\nu}X=Grμ→ν we have d a d a d_(a)d_{a}da smooth G G GGG-monopoles colored by simple roots L e a L e a ^(L)e_(a){ }^{L} e_{a}Lea and otherwise identical. It follows that the common base of SYZ fibrations of X X XXX and Y Y YYY is the configuration space of the smooth monopoles on the real line R R R\mathbb{R}R with n n nnn marked points. The marked points are labeled by the weights μ i μ i mu_(i)\mu_{i}μi of L g L g ^(L)g{ }^{L} \mathrm{~g}L g, which are the singular monopole charges.
An explicit description of Y Y YYY, as well as its category of A-branes D Y D Y D_(Y)\mathscr{D}_{Y}DY, is given [4]. Here we will only describe some of its features. In an open set, Y Y YYY coincides with
Y 0 = a = 1 r k Sym d a A Y 0 = ⨂ a = 1 r k   Sym d a ⁡ A Y_(0)=⨂_(a=1)^(rk)Sym^(d_(a))AY_{0}=\bigotimes_{a=1}^{r k} \operatorname{Sym}^{d_{a}} \mathcal{A}Y0=⨂a=1rkSymda⁡A
the configuration space of d = a = 1 r k d a d = ∑ a = 1 r k   d a d=sum_(a=1)^(rk)d_(a)d=\sum_{a=1}^{r k} d_{a}d=∑a=1rkda points on the punctured Riemann surface A A A\mathcal{A}A, "colored" by simple roots L e a L e a ^(L)e_(a){ }^{L} e_{a}Lea of L g L g ^(L)g{ }^{L} \mathrm{~g}L g, but otherwise identical. The open set is the complement of the divisor of zeros and of poles of function f 0 f 0 f^(0)f^{0}f0 in (5.5).
The top holomorphic form on Y Y YYY is
(5.2) Ω = a = 1 r k α = 1 d a d y α , a y α , a (5.2) Ω = â‹€ a = 1 r k   â‹€ α = 1 d a   d y α , a y α , a {:(5.2)Omega=^^^_(a=1)^(rk)^^^_(alpha=1)^(d_(a))(dy_(alpha,a))/(y_(alpha,a)):}\begin{equation*} \Omega=\bigwedge_{a=1}^{r k} \bigwedge_{\alpha=1}^{d_{a}} \frac{d y_{\alpha, a}}{y_{\alpha, a}} \tag{5.2} \end{equation*}(5.2)Ω=â‹€a=1rk⋀α=1dadyα,ayα,a
where y α , a y α , a y_(alpha,a)y_{\alpha, a}yα,a are coordinates on d d ddd copies of A A A\mathcal{A}A, viewed as the complex plane with 0 and ∞ oo\infty∞ deleted. While Ω Î© Omega\OmegaΩ itself is not globally well defined, so K Y K Y K_(Y)K_{Y}KY is not trivial, Ω 2 Ω ⊗ 2 Omega^(ox2)\Omega^{\otimes 2}Ω⊗2 is well defined and
(5.3) 2 c 1 ( K Y ) = 0 (5.3) 2 c 1 K Y = 0 {:(5.3)2c_(1)(K_(Y))=0:}\begin{equation*} 2 c_{1}\left(K_{Y}\right)=0 \tag{5.3} \end{equation*}(5.3)2c1(KY)=0
This allows D Y D Y D_(Y)\mathscr{D}_{Y}DY to have a Z Z Z\mathbb{Z}Z-valued cohomological grading. The symplectic form on Y Y YYY is inherited from the symplectic form on y y yyy, by restricting it to the vanishing ( S 1 ) d S 1 d (S^(1))^(d)\left(S^{1}\right)^{d}(S1)d in each of its ( C × ) d C × d (C^(xx))^(d)\left(\mathbb{C}^{\times}\right)^{d}(C×)d fibers over Y Y YYY [4]. The precise choice of symplectic structure is the one compatible
with mirror symmetry which we used to define Y Y YYY, as the equivariant mirror of X = G r μ ν X = G r μ → ν X=Gr^( vec(mu))_(nu)\mathcal{X}=\mathrm{Gr}^{\vec{\mu}}{ }_{\nu}X=Grμ→ν and the ordinary mirror of its core.
Including the equivariant T T TTT-equivariant action on X X X\mathcal{X}X and X X XXX corresponds to adding to the σ σ sigma\sigmaσ-model on Y Y YYY a potential
(5.4) W = λ 0 W 0 + a = 1 r k λ a W a (5.4) W = λ 0 W 0 + ∑ a = 1 r k   λ a W a {:(5.4)W=lambda_(0)W^(0)+sum_(a=1)^(rk)lambda_(a)W^(a):}\begin{equation*} W=\lambda_{0} W^{0}+\sum_{a=1}^{r k} \lambda_{a} W^{a} \tag{5.4} \end{equation*}(5.4)W=λ0W0+∑a=1rkλaWa
which is a multivalued holomorphic function on Y ; λ a Y ; λ a Y;lambda_(a)Y ; \lambda_{a}Y;λa are the equivariant parameters of the Λ Î› Lambda\LambdaΛ-action on X X X\mathcal{X}X, and
q = e 2 π i λ 0 q = e 2 Ï€ i λ 0 q=e^(2pi ilambda_(0))q=e^{2 \pi i \lambda_{0}}q=e2Ï€iλ0
The potentials W 0 W 0 W^(0)W^{0}W0 and W a W a W^(a)W^{a}Wa are given by
W 0 = ln f 0 , W a = ln α = 1 d a y a , α W 0 = ln ⁡ f 0 , W a = ln ⁡ ∏ α = 1 d a   y a , α W^(0)=ln f^(0),quadW^(a)=ln prod_(alpha=1)^(d_(a))y_(a,alpha)W^{0}=\ln f^{0}, \quad W^{a}=\ln \prod_{\alpha=1}^{d_{a}} y_{a, \alpha}W0=ln⁡f0,Wa=ln⁡∏α=1daya,α
where
(5.5) f 0 ( y ) = a = 1 r k α = 1 d a i ( 1 a i / y α , a ) L e a , μ i ( b , β ) ( a , α ) ( 1 y β , b / y α , a ) L a , e L e b / 2 (5.5) f 0 ( y ) = ∏ a = 1 r k   ∏ α = 1 d a   ∏ i   1 − a i / y α , a ⟨ L e a , μ i ∏ ( b , β ) ≠ ( a , α )   1 − y β , b / y α , a L a , e L e b / 2 {:(5.5)f^(0)(y)=prod_(a=1)^(rk)prod_(alpha=1)^(d_(a))(prod_(i)(1-a_(i)//y_(alpha,a))^((:L)e_(a),mu_(i):))/(prod_((b,beta)!=(a,alpha))(1-y_(beta,b)//y_(alpha,a))^((:L_(a),e^(L)e_(b):)//2)):}\begin{equation*} f^{0}(y)=\prod_{a=1}^{r k} \prod_{\alpha=1}^{d_{a}} \frac{\left.\prod_{i}\left(1-a_{i} / y_{\alpha, a}\right)^{\langle L} e_{a}, \mu_{i}\right\rangle}{\prod_{(b, \beta) \neq(a, \alpha)}\left(1-y_{\beta, b} / y_{\alpha, a}\right)^{\left\langle L_{a}, e^{L} e_{b}\right\rangle / 2}} \tag{5.5} \end{equation*}(5.5)f0(y)=∏a=1rk∏α=1da∏i(1−ai/yα,a)⟨Lea,μi⟩∏(b,β)≠(a,α)(1−yβ,b/yα,a)⟨La,eLeb⟩/2
The superpotential W W WWW breaks the conformal invariance of the σ σ sigma\sigmaσ-model to Y Y YYY if λ 0 0 λ 0 ≠ 0 lambda_(0)!=0\lambda_{0} \neq 0λ0≠0, since only a quasihomogenous superpotential is compatible with it. This is mirror to breaking of conformal invariance on X X X\mathcal{X}X by the C q × C q × C_(q)^(xx)\mathbb{C}_{q}^{\times}Cq×-action for q 1 q ≠ 1 q!=1q \neq 1q≠1.
Since W 0 W 0 W^(0)W^{0}W0 and W a W a W^(a)W^{a}Wa are multivalued, Y Y YYY is equipped with a collection of closed oneforms with integer periods
c 0 = d W 0 / 2 π i , c a = d W a / 2 π i H 1 ( Y , Z ) c 0 = d W 0 / 2 Ï€ i , c a = d W a / 2 Ï€ i ∈ H 1 ( Y , Z ) c^(0)=dW^(0)//2pi i,quadc^(a)=dW^(a)//2pi i inH^(1)(Y,Z)c^{0}=d W^{0} / 2 \pi i, \quad c^{a}=d W^{a} / 2 \pi i \in H^{1}(Y, \mathbb{Z})c0=dW0/2Ï€i,ca=dWa/2Ï€i∈H1(Y,Z)
which introduce additional gradings in the category of A-branes, as in [73].

5.2.4.

From the mirror perspective, the conformal blocks of L g ^ L g ^ widehat(L_(g))\widehat{L_{\mathrm{g}}}Lg^ come from the B-twisted Landau-Ginsburg model ( Y , W ) ( Y , W ) (Y,W)(Y, W)(Y,W) on D D DDD which is an infinitely long cigar, with A-type boundary condition at infinity corresponding to a Lagrangian L Y L ∈ Y L in YL \in YL∈Y. The partition function of the theory has the following form:
(5.6) V α [ L ] = L Φ α Ω e W (5.6) V α [ L ] = ∫ L   Φ α Ω e − W {:(5.6)V_(alpha)[L]=int_(L)Phi_(alpha)Omegae^(-W):}\begin{equation*} \mathcal{V}_{\alpha}[L]=\int_{L} \Phi_{\alpha} \Omega e^{-W} \tag{5.6} \end{equation*}(5.6)Vα[L]=∫LΦαΩe−W
where Φ α Φ α Phi_(alpha)\Phi_{\alpha}Φα are chiral ring operators, inserted at the tip of the cigar [22,39,40]. By placing the trivial insertion at the origin instead, we get the equivariant central charge function Z [ L ] = Z [ L ] = Z[L]=\mathcal{Z}[L]=Z[L]= L Ω e W ∫ L   Ω e − W int_(L)Omegae^(-W)\int_{L} \Omega e^{-W}∫LΩe−W; by further turning the equivariant parameters off, the potential W W WWW vanishes and the equivariant central charge becomes the ordinary brane central charge Z 0 [ L ] = L Ω Z 0 [ L ] = ∫ L   Ω Z^(0)[L]=int_(L)OmegaZ^{0}[L]=\int_{L} \OmegaZ0[L]=∫LΩ.
We have (re)discovered, from mirror symmetry, an integral representation of the conformal blocks of L g ^ L g ^ widehat(L_(g))\widehat{L_{\mathrm{g}}}Lg^. This "free field representation" of conformal blocks, remarkable for its simplicity [32], goes back to the 1980s work of Kohno and Feigin and Frenkel [34,54], and of Schechtman and Varchenko [69,70].

5.2.5.

There is a reconstruction theory, due to Givental [38] and Teleman [76], which says that, starting with the solution of the quantum differential equation or its mirror counterpart, one gets to reconstruct all genus topological string amplitudes for any semisimple 2D field theory. The semisimplicity condition is satisfied in our case, as W W WWW has isolated critical points. It follows the B-twisted Landau-Ginsburg model on ( Y , W ) ( Y , W ) (Y,W)(Y, W)(Y,W) and A-twisted T T TTT-equivariant sigma model on X X X\mathcal{X}X are equivalent to all genus [2]. Thus, equivariant mirror symmetry holds as an equivalence of topological string amplitudes.

5.3. Equivariant Fukaya-Seidel category

For every A-brane L L LLL at the boundary at infinity of the cigar D D DDD, we get a solution of the K Z K Z KZ\mathrm{KZ}KZ equation. The brane is an object of
D Y = D ( F S ( Y , W ) ) D Y = D ( F S ( Y , W ) ) D_(Y)=D(FS(Y,W))\mathscr{D}_{Y}=D(\mathscr{F} S(Y, W))DY=D(FS(Y,W))
the derived Fukaya-Seidel category of Y Y YYY with potential W W WWW. The category should be thought of as a category of equivariant A-branes, due to the fact W W WWW in (5.4) is multivalued. Another novel aspect of D Y D Y D_(Y)\mathscr{D}_{Y}DY is that it provides an example of Fukaya-Seidel category with coefficients in perverse schobers. This structure, inherited from equivariant mirror symmetry, was discovered in [ 4 ] [ 4 ] [4][4][4].

5.3.1.

Objects of D Y D Y D_(Y)\mathscr{D}_{Y}DY are Lagrangians in Y Y YYY, equipped with some extra data. A Lagrangian in Y Y YYY is a product of d d ddd one-dimensional curves on A A A\mathscr{A}A which are colored by simple roots and may be immersed; or a simplex obtained from an embedded curve, as a configuration space of d d ddd partially ordered colored points. The theory also includes more abstract branes, which are iterated mapping cones over morphisms between Lagrangians.

5.3.2.

The extra data includes a grading by Maslov and equivariant degrees. The equivariant grading of a brane in D Y D Y D_(Y)\mathscr{D}_{Y}DY is defined by choosing a lift of the phase of e W e − W e^(-W)e^{-W}e−W to a real-valued function on the Lagrangian L L LLL. The equivariant degree shift operation,
L L { d } L → L { d → } L rarr L{ vec(d)}L \rightarrow L\{\vec{d}\}L→L{d→}
with d Z r k + 1 d → ∈ Z r k + 1 vec(d)inZ^(rk+1)\vec{d} \in \mathbb{Z}^{r k+1}d→∈Zrk+1, corresponds to changing the lift of W W WWW on L L LLL, now viewed as a graded Lagrangian, W | L { d } = W | L + 2 π i λ d W L { d → } = W L + 2 Ï€ i λ → â‹… d → W|_(L{ vec(d)})=W|_(L)+2pi i vec(lambda)* vec(d)\left.W\right|_{L\{\vec{d}\}}=\left.W\right|_{L}+2 \pi i \vec{\lambda} \cdot \vec{d}W|L{d→}=W|L+2Ï€iλ→⋅d→. This is analogous to how a choice of a lift of the phase of Ω 2 Ω ⊗ 2 Omega^(ox2)\Omega^{\otimes 2}Ω⊗2 defines the Maslov, or cohomological, grading of a Lagrangian. This restricts the Lagrangians that give rise to objects of D Y D Y D_(Y)\mathscr{D}_{Y}DY to those for which such lifts can be defined.
More generally, branes in D Y D Y D_(Y)\mathscr{D}_{Y}DY are graded Lagrangians L L LLL equipped with an extra structure of a local system Λ Î› Lambda\LambdaΛ of modules of a certain algebra B B B\mathscr{B}B we will describe shortly. For the time being, only branes for which Λ Î› Lambda\LambdaΛ is trivial will play a role for us.

5.3.3.

The space of morphisms between a pair of Lagrangian branes in a derived Fukaya category
Hom D Y , ( L 0 , L 1 ) = ker Q / im Q Hom D Y ∗ , ∗ ⁡ L 0 , L 1 = ker ⁡ Q / im ⁡ Q Hom_(D_(Y))^(**,**)(L_(0),L_(1))=ker Q//im Q\operatorname{Hom}_{\mathscr{D}_{Y}}^{*, *}\left(L_{0}, L_{1}\right)=\operatorname{ker} Q / \operatorname{im} QHomDY∗,∗⁡(L0,L1)=ker⁡Q/im⁡Q
is defined by Floer theory, which itself is modeled after Morse theory approach to supersymmetric quantum mechanics, from the introduction. The role of the Morse complex is taken by the Floer complex.
For branes equipped with a trivial local system, the Floer complex
(5.7) CF , ( L 0 , L 1 ) = P L 0 L 1 C P (5.7) CF ∗ , ∗ ⁡ L 0 , L 1 = ⨁ P ∈ L 0 ∩ L 1   C P {:(5.7)CF^(**,**)(L_(0),L_(1))=bigoplus_(PinL_(0)nnL_(1))CP:}\begin{equation*} \operatorname{CF}^{*, *}\left(L_{0}, L_{1}\right)=\bigoplus_{\mathcal{P} \in L_{0} \cap L_{1}} \mathbb{C} \mathcal{P} \tag{5.7} \end{equation*}(5.7)CF∗,∗⁡(L0,L1)=⨁P∈L0∩L1CP
is a graded vector space spanned by the intersection points of the two Lagrangians, together with the action of a differential Q Q QQQ. The complex is graded by the fermion number, which is the Maslov index, and the equivariant gradings, thanks to the fact W W WWW is multivalued.
The action of the differential on this space
Q : C F , ( L 0 , L 1 ) C F + 1 , ( L 0 , L 1 ) Q : C F ∗ , ∗ L 0 , L 1 → C F ∗ + 1 , ∗ L 0 , L 1 Q:CF^(**,**)(L_(0),L_(1))rarrCF^(**+1,**)(L_(0),L_(1))Q: \mathrm{CF}^{*, *}\left(L_{0}, L_{1}\right) \rightarrow \mathrm{CF}^{*+1, *}\left(L_{0}, L_{1}\right)Q:CF∗,∗(L0,L1)→CF∗+1,∗(L0,L1)
is generated by instantons. In Floer theory, the coefficient of P P ′ P^(')\mathcal{P}^{\prime}P′ in Q P Q P QPQ \mathcal{P}QP is obtained by "counting" holomorphic strips in Y Y YYY with boundary on L 0 L 0 L_(0)L_{0}L0 and L 1 L 1 L_(1)L_{1}L1, interpolating from P P P\mathscr{P}P to P P ′ P^(')\mathcal{P}^{\prime}P′, of Maslov index 1 and equivariant degree 0 . The cohomology of the resulting complex is Floer cohomology.

5.3.4.

A simplification in the present case is that, just as branes have a description in terms of the Riemann surface, so do their intersection points, as well as the maps between them.
The theory that results is a generalization of Heegard-Floer theory, which is associated to L g = g l 1 1 L g = g l 1 ∣ 1 ^(L)g=gl_(1∣1){ }^{L} \mathfrak{g}=\mathfrak{g l}_{1 \mid 1}Lg=gl1∣1 and categorifies the Alexander polynomial [63,64]. Heegard-Floer theory

thought of as a configuration space of fermions on the Riemann surface, as opposed to anyons for Y S u 2 Y S u 2 Y_(Su_(2))Y_{\mathfrak{S u}_{2}}YSu2; in particular, their top holomorphic forms differ.
While we so far assumed that L g L g ^(L)g{ }^{L} \mathrm{~g}L g is simply laced, the D Y D Y D_(Y)\mathscr{D}_{Y}DY has an extension to nonsimply-laced Lie algebras, as well as g l m n g l m ∣ n gl_(m∣n)\mathrm{gl}_{m \mid n}glm∣n and s p p m 2 n s p p m ∣ 2 n spp_(m∣2n)\mathfrak{s p} \mathfrak{p}_{m \mid 2 n}sppm∣2n Lie superalgebras, described in [3,5].
Mirror symmetry helps us understand exactly which questions we need to ask to recover homological knot invariants from Y Y YYY.

5.4.1.

Since Y Y YYY is the ordinary mirror of X X XXX, we should start by understanding how to recover homological knot invariants from X X XXX, rather than X X X\mathcal{X}X. Every B-brane on X X X\mathcal{X}X which is relevant for us comes from a B B BBB-brane on X X XXX via an exact functor
(5.8) f : D X D x (5.8) f ∗ : D X → D x {:(5.8)f_(**):D_(X)rarrDx:}\begin{equation*} f_{*}: \mathscr{D}_{X} \rightarrow \mathscr{D} x \tag{5.8} \end{equation*}(5.8)f∗:DX→Dx
which interprets a sheaf "downstairs" on X X XXX as a sheaf "upstairs" on X X X\mathcal{X}X. The functor f f ∗ f_(**)f_{*}f∗ is more precisely the right-derived functor R f R f ∗ Rf_(**)R f_{*}Rf∗. Its adjoint
(5.9) f : D x D X (5.9) f ∗ : D x → D X {:(5.9)f^(**):D_(x)rarrD_(X):}\begin{equation*} f^{*}: \mathscr{D}_{x} \rightarrow \mathscr{D}_{X} \tag{5.9} \end{equation*}(5.9)f∗:Dx→DX
is the left derived functor L f L f ∗ Lf^(**)L f^{*}Lf∗, and corresponds to tensoring with the structure sheaf O X ⊗ O X oxO_(X)\otimes \mathcal{O}_{X}⊗OX, and restricting. Adjointness implies that, given any pair of branes on X X X\mathcal{X}X that come from X X XXX,
F = f F , E = f G F = f ∗ F , E = f ∗ G F=f_(**)F,quadE=f_(**)G\mathscr{F}=f_{*} F, \quad \mathcal{E}=f_{*} GF=f∗F,E=f∗G
the Hom's upstairs, in D X D X D_(X)\mathscr{D}_{X}DX, agree with the Hom's downstairs, in D X D X D_(X)\mathscr{D}_{X}DX,
(5.10) Hom D X X ( F , E ) = Hom D X ( f f F , G ) (5.10) Hom D X X ⁡ ( F , E ) = Hom D X ⁡ f ∗ f ∗ F , G {:(5.10)Hom_(DX)^(X)(F","E)=Hom_(D_(X))(f^(**)f_(**)F,G):}\begin{equation*} \operatorname{Hom}_{\mathscr{D} X}^{X}(\mathcal{F}, \mathscr{E})=\operatorname{Hom}_{\mathscr{D}_{X}}\left(f^{*} f_{*} F, G\right) \tag{5.10} \end{equation*}(5.10)HomDXX⁡(F,E)=HomDX⁡(f∗f∗F,G)
after replacing F F FFF with f f F f ∗ f ∗ F f^(**)f_(**)Ff^{*} f_{*} Ff∗f∗F. The functor f f f ∗ f ∗ f^(**)f_(**)f^{*} f_{*}f∗f∗ is not identity on D X D X D_(X)\mathscr{D}_{X}DX.

5.4.2.

The equivariant homological mirror symmetry relating D X D X D_(X)\mathscr{D}_{X}DX and D Y D Y D_(Y)\mathscr{D}_{Y}DY is not an equivalence of categories, but a correspondence of branes and Hom's which come from a pair of adjoint functors h h ∗ h_(**)h_{*}h∗ and h h ∗ h^(**)h^{*}h∗, inherited from f f ∗ f_(**)f_{*}f∗ and f f ∗ f^(**)f^{*}f∗ via the downstairs homological mirror symmetry:
Alternatively, h h ∗ h^(**)h^{*}h∗ and h h ∗ h_(**)h_{*}h∗ come by composing the upstairs mirror symmetry with a pair of functors k : D y D Y k ∗ : D y → D Y k^(**):Dy rarrD_(Y)k^{*}: \mathscr{D} y \rightarrow \mathscr{D}_{Y}k∗:Dy→DY and k : D Y D y k ∗ : D Y → D y k_(**):D_(Y)rarrDyk_{*}: \mathscr{D}_{Y} \rightarrow \mathscr{D} yk∗:DY→Dy, which are mirror to f f ∗ f^(**)f^{*}f∗ and f f ∗ f_(**)f_{*}f∗. The functors k , k k ∗ , k ∗ k^(**),k_(**)k^{*}, k_{*}k∗,k∗ come from Lagrangian correspondences; their construction is described in joint work with McBreen, Shende, and Zhou [6]. The functor k k ∗ k_(**)k_{*}k∗ amounts to pairing a brane downstairs, with a vanishing torus fiber over it; this is how Figure 6 arises in our model example. The adjoint functors let us recover answers to all interesting questions about X X X\mathcal{X}X from Y Y YYY.

5.4.3.

For any simply laced Lie algebra L g L g ^(L)g{ }^{L} \mathrm{~g}L g, the branes U D X U ∈ D X U inDXU \in \mathscr{D} XU∈DX which serve as cups upstairs are the structure sheaves of (products of) minuscule Grassmannians, as described in Section 4.3.6. They come via the functor h h ∗ h_(**)h_{*}h∗ from branes I U D Y I U ∈ D Y IU inD_(Y)I U \in \mathscr{D}_{Y}IU∈DY downstairs, on Y Y YYY
U = h I U U = h ∗ I U U=h_(**)IUU=h_{*} I UU=h∗IU
which are (products of) generalized intervals. A minuscule Grassmannian G / P i G / P i G//P_(i)G / P_{i}G/Pi is the h h ∗ − h_(**^(-))h_{*^{-}}h∗− image of a brane which is the configuration space of colored points on an interval ending on a pair of punctures on A A A\mathscr{A}A corresponding to representations V i V i V_(i)V_{i}Vi and V i V i ∗ V_(i)^(**)V_{i}^{*}Vi∗. The points are colored by simple positive roots in μ i + μ i = a d a , i L e a μ i + μ i ∗ = ∑ a   d a , i L e a mu_(i)+mu_(i)^(**)=sum_(a)d_(a,i)^(L)e_(a)\mu_{i}+\mu_{i}^{*}=\sum_{a} d_{a, i}{ }^{L} e_{a}μi+μi∗=∑ada,iLea, and ordered in the sequence by which, to obtain the lowest weight μ i μ i ∗ mu_(i)^(**)\mu_{i}^{*}μi∗ in representation V i V i V_(i)V_{i}Vi, we subtract simple positive

FIGURE 7

The cup and cap A-branes corresponding to the defining representation of L g = s u 4 L g = s u 4 ^(L)g=su_(4){ }^{L} \mathfrak{g}=\mathfrak{s u}_{4}Lg=su4, colored by its three simple roots; they are equivariant mirror to a B-brane supported on a P 4 P 4 P^(4)\mathbb{P}^{4}P4 as its structure sheaf.
roots from the highest weight μ i μ i mu_(i)\mu_{i}μi. Because V i V i V_(i)V_{i}Vi and V i V i ∗ V_(i)^(**)V_{i}^{*}Vi∗ are minuscule representations, the ordering and hence the brane I U I U IUI UIU is unique, up to equivalence and a choice of grading. The U U U\mathcal{U}U branes project back down as generalized figure-eight branes; these are nested products of figure-eights, colored by simple roots
h U = h h I U = E U h ∗ U = h ∗ h ∗ I U = E U h^(**)U=h^(**)h_(**)IU=EUh^{*} U=h^{*} h_{*} I U=E Uh∗U=h∗h∗IU=EU
and ordered analogously, as in Figure 7. As objects of D Y D Y D_(Y)\mathscr{D}_{Y}DY, these branes are best described iterated cones over more elementary branes, mirror to stable basis branes [5]. The cup and cap branes all come with trivial local systems, for which the Floer complexes are the familiar ones, given by (5.7).
As an example, for L g = S H 2 L g = S H 2 ^(L)g=SH_(2){ }^{L} \mathfrak{g}=\mathfrak{S H}_{2}Lg=SH2 the only minuscule representation is the defining representation V i = V 1 2 V i = V 1 2 V_(i)=V_((1)/(2))V_{i}=V_{\frac{1}{2}}Vi=V12, which is self-conjugate. The cup brane U U UUU in X X X\mathcal{X}X is a product of d d ddd non-intersecting P 1 P 1 P^(1)\mathbb{P}^{1}P1, s. It comes, as the image of h h ∗ h_(**)h_{*}h∗, from a brane I U I U IUI UIU in Y Y YYY which is a product of d d ddd simple intervals, connecting pairs of punctures that come together. The U U U\mathcal{U}U-brane projects back down, via the h h ∗ h^(**)h^{*}h∗ functor, as a product of d d ddd elementary figure-eight branes. The branes are graded by Maslov and equivariant gradings, as described in [2].

5.4.4.

In the description based on Y Y YYY, both the branes, and the action of braiding on them is geometric, so we can simply start with a link and a choice of projection to the surface A = R × S 1 A = R × S 1 A=RxxS^(1)\mathcal{A}=\mathbb{R} \times S^{1}A=R×S1. A link contained in a three ball in R 2 × S 1 R 2 × S 1 R^(2)xxS^(1)\mathbb{R}^{2} \times S^{1}R2×S1 is equivalent to the same link in R 3 R 3 R^(3)\mathbb{R}^{3}R3, and projects to a contractible patch on A A A\mathcal{A}A.
To translate the link to a pair of A-branes, start by choosing bicoloring of the link projection, such that each of its components has an equal number of red and blue segments, and the red always underpass the blue. For a link component colored by a representation V i V i V_(i)V_{i}Vi of L g L g ^(L)g{ }^{L} \mathrm{~g}L g, place a puncture colored by its highest weight μ i μ i mu_(i)\mu_{i}μi where a blue segment begins and its conjugate μ i μ i ∗ mu_(i)^(**)\mu_{i}^{*}μi∗ where it ends; the orientation of the link component distinguishes the two. The mirror Lagrangians I u I u I_(u)I_{u}Iu and B E U B E U BE_(U)\mathscr{B} E_{\mathcal{U}}BEU are obtained by replacing all the blue segments by the interval branes, and all the red segments by figure-eight branes, related by equivariant mirror symmetry to minuscule Grassmannian branes. This data determines both Y Y YYY and the branes on it we need. The variant of the second step, applicable for Lie superalgebras, is described in [5].

FIGURE 8

A bicoloring of the left-handed trefoil.
Equivariant mirror symmetry predicts that a homological link invariant is the space of morphisms
(5.11) Hom D Y , ( B E U , I u ) = k Z , d Z r k + 1 Hom D Y ( B E U , I U [ k ] { d } ) (5.11) Hom D Y ∗ , ∗ ⁡ B E U , I u = ⨁ k ∈ Z , d → ∈ Z r k + 1   Hom D Y ⁡ B E U , I U [ k ] { d → } {:(5.11)Hom_(D_(Y))^(**,**)(BE_(U),I_(u))=bigoplus_(k inZ, vec(d)inZ^(rk+1))Hom_(D_(Y))(BE_(U),IU[k]{( vec(d))}):}\begin{equation*} \operatorname{Hom}_{\mathscr{D}_{Y}}^{*, *}\left(\mathscr{B} E_{\mathcal{U}}, I_{u}\right)=\bigoplus_{k \in \mathbb{Z}, \vec{d} \in \mathbb{Z}^{r k+1}} \operatorname{Hom}_{\mathscr{D}_{Y}}\left(\mathscr{B} E_{U}, I U[k]\{\vec{d}\}\right) \tag{5.11} \end{equation*}(5.11)HomDY∗,∗⁡(BEU,Iu)=⨁k∈Z,d→∈Zrk+1HomDY⁡(BEU,IU[k]{d→})
the cohomology of the Floer complex of the two branes. In what follows, will explain how to compute it.

FIGURE 9

The branes corresponding to the left-handed trefoil in L g = s u 2 L g = s u 2 ^(L)g=su_(2){ }^{L} \mathfrak{g}=\mathfrak{s u}_{2}Lg=su2. The knot was isotoped relative to Figure 8 .

5.4.5.

To evaluate the Euler characteristic of the homology in (5.11), one simply counts intersection points of Lagrangians, keeping track of gradings. For links in R 3 R 3 R^(3)\mathbb{R}^{3}R3, the equivariant grading in (5.11) collapses to a Z Z Z\mathbb{Z}Z-grading. The Euler characteristic becomes
(5.12) χ ( B E u , I u ) = P B E U I U ( 1 ) M ( P ) q J ( P ) (5.12) χ ( B E u , I u ) = ⨁ P ∈ B E U ∩ I U   ( − 1 ) M ( P ) q J ( P ) {:(5.12)chi(BEu","Iu)=bigoplus_(PinBEU nn IU)(-1)^(M(P))q^(J(P)):}\begin{equation*} \chi(\mathscr{B} E u, I u)=\bigoplus_{\mathcal{P} \in \mathscr{B} E \mathcal{U \cap I U}}(-1)^{M(\mathcal{P})} q^{J(\mathcal{P})} \tag{5.12} \end{equation*}(5.12)χ(BEu,Iu)=⨁P∈BEU∩IU(−1)M(P)qJ(P)
where M ( P ) M ( P ) M(P)M(\mathscr{P})M(P) and J ( P ) J ( P ) J(P)J(\mathscr{P})J(P) are the Maslov and c 0 c 0 c^(0)c^{0}c0-grading of the point P P P\mathscr{P}P; as in Heegard-Floer theory, there are purely combinatorial formulas for them [3,5]. Mirror symmetry implies that this is the U q ( L g ) U q L g U_(q)(^(L)(g))U_{\mathfrak{q}}\left({ }^{L} \mathrm{~g}\right)Uq(L g) invariant of the link in R 3 R 3 R^(3)\mathbb{R}^{3}R3.
The fact that for L g = s u 2 L g = s u 2 ^(L)g=su_(2){ }^{L} \mathfrak{g}=\mathfrak{s u}_{2}Lg=su2 the graded count of intersection points in (5.12) reproduces the Jones polynomial is a theorem of Bigelow [15], building on the work of Lawrence [56-58]. Bigelow also proved the statement for L g = s u N L g = s u N ^(L)g=su_(N){ }^{L} \mathfrak{g}=\mathfrak{s u}_{N}Lg=suN with links colored by the defining representation [16]. The equivariant homological mirror symmetry explains the origin of Bigelow's peculiar construction, and generalizes it to other U q ( L g ) U q L g U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(L g) link invariants. 1 1 ^(1){ }^{1}1

5.4.6.

The action of the differential Q Q QQQ on the Floer complex, defined by counting holomorphic maps from a disk D D DDD to Y Y YYY with boundaries on the pair of Lagrangians, should have a reformulation [2] in terms of counting holomorphic curves embedded in D × A D × A D xxAD \times \mathcal{A}D×A with certain properties, generalizing the cylindrical formulation of Heegard-Floer theory due to Lipshitz [59]. The curve must have a projection to D D DDD as a d = a d a d = ∑ a   d a d=sum_(a)d_(a)d=\sum_{a} d_{a}d=∑ada-fold cover, with branching only between components of one color, and a projection to A A A\mathscr{A}A as a domain with boundaries on one-dimensional Lagrangians of matching colors. In addition, the potential W W WWW must pull back to D D DDD as a regular holomorphic function. Computing the action of Q Q QQQ in this framework reduces to solving a sequence of well defined, but hard, problems in complex analysis in one dimension, which are applications of the Riemannian mapping theorem, similar to that in [63].
To compute the link homology groups
(6.1) Hom D Y , ( B E U , I U ) (6.1) Hom D Y ∗ , ∗ ⁡ B E U , I U {:(6.1)Hom_(D_(Y))^(**,**)(BE_(U),I_(U)):}\begin{equation*} \operatorname{Hom}_{\mathscr{D}_{Y}}^{*, *}\left(\mathscr{B} E_{\mathcal{U}}, I_{U}\right) \tag{6.1} \end{equation*}(6.1)HomDY∗,∗⁡(BEU,IU)
we will make use of mirror symmetry which relates X X XXX and Y Y YYY and is the equivalence of categories
(6.2) D X D A D Y (6.2) D X ≅ D A ≅ D Y {:(6.2)D_(X)~=D_(A)~=D_(Y):}\begin{equation*} \mathscr{D}_{X} \cong \mathscr{D}_{A} \cong \mathscr{D}_{Y} \tag{6.2} \end{equation*}(6.2)DX≅DA≅DY
proven in [4]. A basic virtue of mirror symmetry is that it sums up holomorphic curve counts. In our case, it solves all the disk-counting problems required to find the action of the differential Q Q QQQ on the Floer complex underlying (6.1).

6.1. The algebra of A-branes

As in the simplest examples of homological mirror symmetry, D X D X D_(X)\mathscr{D}_{X}DX and D Y D Y D_(Y)\mathscr{D}_{Y}DY are both generated by a finite set of branes.

6.1.1.

From perspective of Y Y YYY, the generating set of branes come from products of real line Lagrangians on A A A\mathscr{A}A, colored by d = a d a d = ∑ a   d a d=sum_(a)d_(a)d=\sum_{a} d_{a}d=∑ada simple roots. The brane is unchanged by reorder ing a pair of its neighboring Lagrangian components, provided they are colored by roots
1 In [60], Bigelow's representation of the Jones polynomial, specialized to q = 1 q = 1 q=1q=1q=1, was related to the Euler characteristic of symplectic Khovanov homology of [73].
which are not linked in the Dynkin diagram L e a , L e b = 0 L e a , L e b = 0 (:^(L)e_(a),^(L)e_(b):)=0\left\langle{ }^{L} e_{a},{ }^{L} e_{b}\right\rangle=0⟨Lea,Leb⟩=0. It is also unchanged by passing a component colored by L e a L e a ^(L)e_(a){ }^{L} e_{a}Lea across a puncture colored by a weight μ i μ i mu_(i)\mu_{i}μi with L e a , μ i = 0 L e a , μ i = 0 (:^(L)e_(a),mu_(i):)=0\left\langle{ }^{L} e_{a}, \mu_{i}\right\rangle=0⟨Lea,μi⟩=0. There is a generating brane
T e = T i 1 × × T i d D Y T e = T i 1 × ⋯ × T i d ∈ D Y T_(e)=T_(i_(1))xx cdots xxT_(i_(d))inD_(Y)T_{e}=T_{i_{1}} \times \cdots \times T_{i_{d}} \in \mathscr{D}_{Y}Te=Ti1×⋯×Tid∈DY
for every inequivalent ordering of d d ddd colored real lines on the cylinder. Their direct sum
T = T D Y T = ⨁ と   T と ∈ D Y T=bigoplus_(と)T_(と)inD_(Y)T=\bigoplus_{と} T_{と} \in \mathscr{D}_{Y}T=⨁とTと∈DY
is the generator of D Y D Y D_(Y)\mathscr{D}_{Y}DY which is mirror to the tilting vector bundle on X X XXX, which generates D X D X D_(X)\mathscr{D}_{X}DX. This generalizes the simplest example of mirror symmetry from Section 5.1. As before, we will be denoting branes on X X XXX and on Y Y YYY related by homological mirror symmetry by the same letter.

6.1.2.

A well known phenomenon in mirror symmetry is that it may introduce Lagrangians with an extra structure of a local system, a nontrivial flat U ( 1 ) U ( 1 ) U(1)U(1)U(1) bundle. The mirror of a structure sheaf of a generic point, in our model example of mirror symmetry from Section 5.1, is a Lagrangian of this sort.
Here, we find a generalization of this [4]. The pair of adjoint functors h h ∗ h_(**)h_{*}h∗ and h h ∗ h^(**)h^{*}h∗ that relate D Y D Y D_(Y)\mathscr{D}_{Y}DY with its equivariant mirror D X D X D_(X)\mathscr{D}_{X}DX equip each T T TTT-brane with a vector bundle or, more precisely, with a local system of modules for a graded algebra B B B\mathscr{B}B. The algebra is a product B = a = 1 r k B d a B = ⨂ a = 1 r k   B d a B=⨂_(a=1)^(rk)B_(d_(a))\mathscr{B}=\bigotimes_{a=1}^{r k} \mathscr{B}_{d_{a}}B=⨂a=1rkBda, where B d B d B_(d)\mathscr{B}_{d}Bd has a representation as the quotient of the algebra of polynomials in d d ddd variables z 1 , , z d z 1 , … , z d z_(1),dots,z_(d)z_{1}, \ldots, z_{d}z1,…,zd which sets their symmetric functions to zero. The z z zzz 's have equivariant q q qqq-degree equal to one.
As a consequence, the downstairs theory is not simply the Fukaya category of Y Y YYY, but a Fukaya category with coefficients [4]: Floer complexes assign to each intersection point P L 0 L 1 P ∈ L 0 ∩ L 1 PinL_(0)nnL_(1)\mathscr{P} \in L_{0} \cap L_{1}P∈L0∩L1 a vector space hom B ( Λ 0 | P , Λ 1 | P ) B Λ 0 P , Λ 1 P _(B)(Lambda_(0)|P,Lambda_(1)|P)_{\mathscr{B}}\left(\Lambda_{0}\left|\mathcal{P}, \Lambda_{1}\right| \mathcal{P}\right)B(Λ0|P,Λ1|P) of homomorphisms of B B B\mathscr{B}B-modules Λ 0 , 1 Λ 0 , 1 Lambda_(0,1)\Lambda_{0,1}Λ0,1 which L 0 , 1 L 0 , 1 L_(0,1)L_{0,1}L0,1 are equipped with. The cup and cap branes E U E U E_(U)E_{\mathcal{U}}EU and I U I U IUI \mathcal{U}IU come with trivial modules for B B B\mathscr{B}B. The T C T C T_(C)T_{\mathscr{C}}TC branes correspond to modules for B B B\mathscr{B}B equal to B B B\mathscr{B}B itself.

6.1.3.

Since the T C T C T_(C)T_{\mathcal{C}}TC-branes are noncompact, defining the Hom's between them requires care. The Hom's
Hom D Y ( T , T [ k ] { d } ) = HF ( T ζ , T [ k ] { d } ) Hom D Y ⁡ T ⨀ , T ⨀ ′ [ k ] { d → } = HF ⁡ T ⨀ ζ , T ⨀ ′ [ k ] { d → } Hom_(D_(Y))(T_(⨀),T_(⨀)^(')[k]{( vec(d))})=HF(T_(⨀)^(zeta),T_(⨀)^(')[k]{( vec(d))})\operatorname{Hom}_{\mathscr{D}_{Y}}\left(T_{\bigodot}, T_{\bigodot}^{\prime}[k]\{\vec{d}\}\right)=\operatorname{HF}\left(T_{\bigodot}^{\zeta}, T_{\bigodot}^{\prime}[k]\{\vec{d}\}\right)HomDY⁡(T⨀,T⨀′[k]{d→})=HF⁡(T⨀ζ,T⨀′[k]{d→})
are defined through a perturbation of T C T C T_(C)T_{\mathcal{C}}TC which induces wrapping near infinities of A A A\mathcal{A}A, as in Figure 4, and other examples of wrapped Fukaya categories.
The Floer cohomology groups HF are cohomology groups of the Floer complex whose generators are intersection points of the T e T e T_(e)T_{e}Te branes, with coefficients in B B B\mathscr{B}B. The generators all have homological degree zero, so the Floer differential is trivial, and
(6.3) Hom D Y ( T e , T e [ k ] { d } ) = 0 , for all k 0 and all d (6.3) Hom D Y ⁡ T e , T e ′ [ k ] { d → } = 0 ,  for all  k ≠ 0  and all  d → {:(6.3)Hom_(D_(Y))(T_(e),T_(e^('))[k]{( vec(d))})=0","quad" for all "k!=0" and all " vec(d):}\begin{equation*} \operatorname{Hom}_{\mathscr{D}_{Y}}\left(T_{e}, T_{\mathcal{e}^{\prime}}[k]\{\vec{d}\}\right)=0, \quad \text { for all } k \neq 0 \text { and all } \vec{d} \tag{6.3} \end{equation*}(6.3)HomDY⁡(Te,Te′[k]{d→})=0, for all k≠0 and all d→
The Floer product on D Y D Y D_(Y)\mathscr{D}_{Y}DY makes
A = Hom D Y ( T , T ) = Υ , C d Z r k + 1 Hom D X ( T e , T e { d } ) A = Hom D Y ∗ ⁡ ( T , T ) = ⨁ Î¥ , C ′   ⨁ d → ∈ Z r k + 1   Hom D X ⁡ T e , T e ′ { d → } A=Hom_(D_(Y))^(**)(T,T)=bigoplus_(Î¥,C^('))bigoplus_( vec(d)inZ^(rk+1))Hom_(D_(X))(T_(e),T_(e^(')){( vec(d))})A=\operatorname{Hom}_{\mathscr{D}_{Y}}^{*}(T, T)=\bigoplus_{\Upsilon, \mathscr{C}^{\prime}} \bigoplus_{\vec{d} \in \mathbb{Z}^{r k+1}} \operatorname{Hom}_{\mathscr{D}_{X}}\left(T_{e}, T_{\mathcal{e}^{\prime}}\{\vec{d}\}\right)A=HomDY∗⁡(T,T)=⨁Υ,C′⨁d→∈Zrk+1HomDX⁡(Te,Te′{d→})
into an algebra, which is an ordinary associative algebra, graded only by equivariant degrees.

6.1.4.

The vanishing in (6.3) mirrors the tilting property of T T TTT viewed as the generator of D X D X D_(X)\mathscr{D}_{X}DX. The tilting vector bundle T D X T ∈ D X T inD_(X)T \in \mathscr{D}_{X}T∈DX is inherited from the Bezrukavnikov-Kaledin tilting bundle T T T\mathcal{T}T on X X X\mathcal{X}X,
T = C T e D x T = ⨁ C   T e ∈ D x T=bigoplus_(C)T_(e)inDx\mathcal{T}=\bigoplus_{\mathcal{C}} \mathcal{T}_{\mathcal{e}} \in \mathscr{D} xT=⨁CTe∈Dx
from Section 4.4, as the image of the f f ∗ f^(**)f^{*}f∗ functor, which is tensoring with the structure sheaf of X X XXX and restriction, f T = T D X f ∗ T = T ∈ D X f^(**)T=T inD_(X)f^{*} \mathcal{T}=T \in \mathscr{D}_{X}f∗T=T∈DX. The endomorphism of the upstairs tilting generator T ,
A = Hom D X ( T , T ) A = Hom D X ∗ ⁡ ( T , T ) A=Hom_(DX)^(**)(T,T)\mathscr{A}=\operatorname{Hom}_{\mathscr{D} X}^{*}(\mathcal{T}, \mathcal{T})A=HomDX∗⁡(T,T)
is the cylindrical KLRW algebra.
Since T T T\mathcal{T}T is a vector bundle on X X X\mathcal{X}X, the center of A A A\mathscr{A}A is the algebra of holomorphic functions on X X X\mathcal{X}X. The downstairs algebra is a quotient of the upstairs one, by a two-sided ideal
(6.4) A = A / I (6.4) A = A / I {:(6.4)A=A//I:}\begin{equation*} A=\mathscr{A} / \mathscr{I} \tag{6.4} \end{equation*}(6.4)A=A/I
The ideal I I I\mathscr{I}I is generated by holomorphic functions that vanish on the core X X XXX.

6.1.5.

The cKLRW algebra A A A\mathscr{A}A is defined as an algebra of colored strands on a cylinder, decorated with dots, where composition is represented by stacking cylinders and rescaling [80]. The local algebra relations are those of the ordinary KLRW algebra from [78]. Placing the theory on the cylinder, it gets additional gradings by the winding number of strands of a given color, corresponding to the equivariant Λ Î› Lambda\LambdaΛ-action on X X X\mathcal{X}X.
The elements of the algebra A = A / I A = A / I A=A//IA=\mathscr{A} / \mathscr{I}A=A/I have a geometric interpretation by recalling the Floer complex C F ( T C , T ) C F ∗ T C , T ⨀ ′   CF^(**)(T_(C),T_(⨀'))\mathrm{CF}^{*}\left(T_{\mathcal{C}}, T_{\bigodot^{\prime}}\right)CF∗(TC,T⨀′) is fundamentally generated by paths rather the intersection points. The S 1 S 1 S^(1)S^{1}S1 of the algebra cylinder is the S 1 S 1 S^(1)S^{1}S1 in the Riemann surface A A A\mathcal{A}A; its vertical direction parameterizes the path. The geometric intersection points of the T T TTT-branes on A A A\mathscr{A}A correspond to strings in A A AAA. The flat bundle morphisms, a copy of B B B\mathscr{B}B for each geometric intersection point, dress the strings by dots of the same color. The algebra B B B\mathscr{B}B is the quotient, of the subalgebra of A A A\mathscr{A}A generated by dots, by the ideal I I I\mathscr{I}I of their symmetric functions.
Since T = T e T = ⨁ と   T e T=bigoplus_(と)T_(e)T=\bigoplus_{と} T_{\mathscr{e}}T=⨁とTe generates D Y D Y D_(Y)\mathscr{D}_{Y}DY, like every Lagrangian in D Y D Y D_(Y)\mathscr{D}_{Y}DY, the B E U B E U BEU\mathscr{B} E \mathcal{U}BEU brane has a description as a complex
(6.5) B E u t 1 B E 1 ( T ) t 0 B E 0 ( T ) , (6.5) B E u ≅ ⋯ → t 1 B E 1 ( T ) → t 0 B E 0 ( T ) , {:(6.5)BE_(u)~=cdotsrarr"t_(1)"BE_(1)(T)rarr"t_(0)"BE_(0)(T)",":}\begin{equation*} \mathscr{B} E_{u} \cong \cdots \xrightarrow{t_{1}} \mathscr{B} E_{1}(T) \xrightarrow{t_{0}} \mathscr{B} E_{0}(T), \tag{6.5} \end{equation*}(6.5)BEu≅⋯→t1BE1(T)→t0BE0(T),
every term of which is a direct sum of T C T C T_(C)T_{\mathscr{C}}TC-branes. The complex is the projective resolution of the B E U B E U BE_(U)\mathscr{B} E_{U}BEU brane. It describes how to get the B E U D Y B E U ∈ D Y BE_(U)inD_(Y)\mathscr{B} E_{U} \in \mathscr{D}_{Y}BEU∈DY brane by starting with the direct sum brane
(6.6) B E ( T ) = k B E k ( T ) [ k ] (6.6) B E ( T ) = ⨁ k   B E k ( T ) [ k ] {:(6.6)BE(T)=bigoplus_(k)BE_(k)(T)[k]:}\begin{equation*} \mathscr{B} E(T)=\bigoplus_{k} \mathscr{B} E_{k}(T)[k] \tag{6.6} \end{equation*}(6.6)BE(T)=⨁kBEk(T)[k]
with a trivial differential, and taking iterated cones over elements t k A t k ∈ A t_(k)in At_{k} \in Atk∈A. This deforms the differential to
(6.7) Q A = k t k A (6.7) Q A = ∑ k   t k ∈ A {:(6.7)Q_(A)=sum_(k)t_(k)in A:}\begin{equation*} Q_{A}=\sum_{k} t_{k} \in A \tag{6.7} \end{equation*}(6.7)QA=∑ktk∈A
which takes
Q A : B E ( T ) B E ( T ) [ 1 ] Q A : B E ( T ) → B E ( T ) [ 1 ] Q_(A):BE(T)rarrBE(T)[1]Q_{A}: \mathscr{B} E(T) \rightarrow \mathscr{B} E(T)[1]QA:BE(T)→BE(T)[1]
as a cohomological degree 1 and equivariant degree 0 operator, which squares to zero Q A 2 = 0 Q A 2 = 0 Q_(A)^(2)=0Q_{A}^{2}=0QA2=0 in the algebra A A AAA.

6.2.1.

The category of A-branes D Y D Y D_(Y)\mathscr{D}_{Y}DY has a second, Koszul dual set of generators, which are the vanishing cycle branes I = I I = ⨁ ↼   I ↼ I=bigoplus_(↼)I_(↼)I=\bigoplus_{\leftharpoonup} I_{\leftharpoonup}I=⨁↼I↼ of [2]. The I I III - and the T T TTT-branes are dual in the sense that the only nonvanishing morphisms from the T T TTT - to the I I III-branes are
(6.8) Hom D Y ( T φ , I φ ) = C δ φ , φ (6.8) Hom D Y ⁡ T φ , I φ ′ = C δ φ , φ ′ {:(6.8)Hom_(D_(Y))(T_(varphi),I_(varphi^(')))=Cdelta_(varphi,varphi^(')):}\begin{equation*} \operatorname{Hom}_{\mathscr{D}_{Y}}\left(T_{\varphi}, I_{\varphi^{\prime}}\right)=\mathbb{C} \delta_{\varphi, \varphi^{\prime}} \tag{6.8} \end{equation*}(6.8)HomDY⁡(Tφ,Iφ′)=Cδφ,φ′
The I C I C I_(C)I_{\mathscr{C}}IC-branes and the T C T C TCT \mathcal{C}TC-branes are, respectively, the simple and the projective modules of the algebra A A AAA.

6.2.2.

Among the I I III-branes, we find the branes I u D Y I u ∈ D Y Iu inD_(Y)I u \in \mathscr{D}_{Y}Iu∈DY which serve as cups. This is a wonderful simplification because it implies that from the complex in (6.5), we get for free a complex of vector spaces:
(6.9) 0 hom A ( B E 0 ( T ) , I u { d } ) t 0 hom A ( B E 1 ( T ) , I u { d } ) t 1 (6.9) 0 → hom A ⁡ B E 0 ( T ) , I u { d → } → t 0 hom A ⁡ B E 1 ( T ) , I u { d → } → t 1 ⋯ {:(6.9)0rarrhom_(A)(BE_(0)(T),Iu{( vec(d))})rarr"t_(0)"hom_(A)(BE_(1)(T),Iu{( vec(d))})rarr"t_(1)"cdots:}\begin{equation*} 0 \rightarrow \operatorname{hom}_{A}\left(\mathscr{B} E_{0}(T), I u\{\vec{d}\}\right) \xrightarrow{t_{0}} \operatorname{hom}_{A}\left(\mathscr{B} E_{1}(T), I u\{\vec{d}\}\right) \xrightarrow{t_{1}} \cdots \tag{6.9} \end{equation*}(6.9)0→homA⁡(BE0(T),Iu{d→})→t0homA⁡(BE1(T),Iu{d→})→t1⋯
The maps in the complex (6.9) are induced from the complex in (6.5). The cohomologies of this complex are the link homologies we are after,
(6.10) Hom D Y ( B E U , I u [ k ] { d } ) = H k ( hom A ( B E U , I u ) ) (6.10) Hom D Y ⁡ B E U , I u [ k ] { d → } = H k hom A ⁡ B E U , I u {:(6.10)Hom_(D)^(Y)(BE_(U),I_(u)[k]{( vec(d))})=H^(k)(hom_(A)(BE_(U),I_(u))):}\begin{equation*} \operatorname{Hom}_{\mathscr{D}}^{Y}\left(\mathscr{B} E_{\mathcal{U}}, I_{u}[k]\{\vec{d}\}\right)=H^{k}\left(\operatorname{hom}_{A}\left(\mathscr{B} E_{\mathcal{U}}, I_{u}\right)\right) \tag{6.10} \end{equation*}(6.10)HomDY⁡(BEU,Iu[k]{d→})=Hk(homA⁡(BEU,Iu))

6.2.3.

We learn that link homology captures only a small part of the geometry of B E U B E U BEU\mathscr{B} E \mathcal{U}BEU, the braided cup brane, or more precisely, of the complex that resolves it. Because the T T TTT branes are dual to the I I III-branes by (6.8), almost all terms in the complex (6.9) vanish. The cohomology (6.10) of small complex that remains is the U q ( L g ) U q L g U_(q)(^(L)(g))U_{\mathrm{q}}\left({ }^{L} \mathrm{~g}\right)Uq(L g) link homology.

6.2.4.

The complex (6.9) itself has a geometric interpretation as the Floer complex,
C F , ( B E U , I U ) C F ∗ , ∗ ( B E U , I U ) CF^(**,**)(BEU,IU)\mathrm{CF}^{*, *}(\mathscr{B} E \mathcal{U}, I U)CF∗,∗(BEU,IU)
Namely, the vector space at the k k kkk th term of the complex
hom A ( B E k ( T ) , Iu { d } ) hom A ⁡ B E k ( T ) , Iu ⁡ { d → } hom_(A)(BE_(k)(T),Iu{( vec(d))})\operatorname{hom}_{A}\left(\mathscr{B} E_{k}(T), \operatorname{Iu}\{\vec{d}\}\right)homA⁡(BEk(T),Iu⁡{d→})
is identified, by construction described in section 6.3, with that spanned by the intersection points of the B E u B E u BEu\mathscr{B} E uBEu brane and the I U I U IUI UIU brane, of cohomological degree [ k ] [ k ] [k][k][k] and equivariant degree { d } { d → } { vec(d)}\{\vec{d}\}{d→}.
The maps in the complex
t k 1 hom A ( B E k ( T ) , Iu u { d } ) t k hom A ( B E k + 1 ( T ) , I u { d } ) t k + 1 ⋯ → t k − 1 hom A ⁡ B E k ( T ) , Iu ⁡ u { d → } → t k hom A ⁡ B E k + 1 ( T ) , I u { d → } → t k + 1 ⋯ cdotsrarr"t_(k-1)"hom_(A)(BE_(k)(T),Iu u{( vec(d))})rarr"t_(k)"hom_(A)(BE_(k+1)(T),Iu{( vec(d))})rarr"t_(k+1)"cdots\cdots \xrightarrow{t_{k-1}} \operatorname{hom}_{A}\left(\mathscr{B} E_{k}(T), \operatorname{Iu} u\{\vec{d}\}\right) \xrightarrow{t_{k}} \operatorname{hom}_{A}\left(\mathscr{B} E_{k+1}(T), I u\{\vec{d}\}\right) \xrightarrow{t_{k+1}} \cdots⋯→tk−1homA⁡(BEk(T),Iu⁡u{d→})→tkhomA⁡(BEk+1(T),Iu{d→})→tk+1⋯
encode the action of the Floer differential. A priori, computing these requires counting holomorphic disk instantons. In our case, mirror symmetry (6.2) has summed them up.

6.3. Projective resolutions from geometry

The projective resolution in (6.5) encodes all the U q ( L g ) U q L g U_(q)(^(L)g)U_{\mathcal{q}}\left({ }^{L} \mathfrak{g}\right)Uq(Lg) link homology, and more. Finding the resolution requires solving two problems, both in general difficult. We will solve simultaneously [ 5 ] [ 5 ] [5][5][5].

6.3.1.

The first problem is to compute which module of the algebra A A AAA the brane B E u B E u BEu\mathscr{B} E uBEu gets mapped to by the Yoneda functor
L D Y Hom D Y , ( T , L ) D A L ∈ D Y → Hom D Y ∗ , ∗ ⁡ ( T , L ) ∈ D A L inD_(Y)rarrHom_(D_(Y))^(**,**)(T,L)inD_(A)L \in \mathscr{D}_{Y} \rightarrow \operatorname{Hom}_{\mathscr{D}_{Y}}^{*, *}(T, L) \in \mathscr{D}_{A}L∈DY→HomDY∗,∗⁡(T,L)∈DA
This functor, which is the source of the equivalence D Y D A D Y ≅ D A D_(Y)~=D_(A)\mathscr{D}_{Y} \cong \mathscr{D}_{A}DY≅DA, maps a brane L L LLL to a right module for A A AAA, on which the algebra acts as
Hom D Y , ( T , L ) Hom D Y ( T , T ) Hom D Y , ( T , L ) Hom D Y ∗ , ∗ ⁡ ( T , L ) ⊗ Hom D Y ∗ ⁡ ( T , T ) → Hom D Y ∗ , ∗ ⁡ ( T , L ) Hom_(D_(Y))^(**,**)(T,L)oxHom_(D_(Y))^(**)(T,T)rarrHom_(D_(Y))^(**,**)(T,L)\operatorname{Hom}_{\mathscr{D}_{Y}}^{*, *}(T, L) \otimes \operatorname{Hom}_{\mathscr{D}_{Y}}^{*}(T, T) \rightarrow \operatorname{Hom}_{\mathscr{D}_{Y}}^{*, *}(T, L)HomDY∗,∗⁡(T,L)⊗HomDY∗⁡(T,T)→HomDY∗,∗⁡(T,L)
Evaluating this action requires counting disk instantons.

6.3.2.

The second problem is to find the resolution of this module, as in (6.5). The Yoneda functor maps the T e T e T_(e)T_{e}Te branes to projective modules of the algebra A A AAA, so the resolution in (6.5) is a projective resolution of the A A AAA module corresponding to the B E U B E U BE_(U)\mathscr{B} E_{\mathcal{U}}BEU brane. This problem is known to be solvable, however, only formally so, by infinite bar resolutions.

6.3.3.

In our setting, these two problems get solved together. Fortune smiles since the B E U D Y B E U ∈ D Y BE_(U)inD_(Y)\mathscr{B} E_{\mathcal{U}} \in \mathscr{D}_{Y}BEU∈DY branes are products of d d ddd one-dimensional Lagrangians on A A A\mathcal{A}A, for which the complex resolving brane (6.5) can be deduced explicitly, from the geometry of the brane.

6.3.4.

Take a pair of branes L L ′ L^(')L^{\prime}L′ and L L ′ ′ L^('')L^{\prime \prime}L′′ on Y Y YYY which are products of d d ddd one-dimensional Lagrangians on A A A\mathcal{A}A, chosen to coincide up to one of their factors. Up to permutation, we can take
L = L 1 × L 2 × × L d , L = L 1 × L 2 × × L d L ′ = L 1 × L 2 × ⋯ × L d , L ′ ′ = L 1 ′ ′ × L 2 × ⋯ × L d L^(')=L_(1)xxL_(2)xx cdots xxL_(d),quadL^('')=L_(1)^('')xxL_(2)xx cdots xxL_(d)L^{\prime}=L_{1} \times L_{2} \times \cdots \times L_{d}, \quad L^{\prime \prime}=L_{1}^{\prime \prime} \times L_{2} \times \cdots \times L_{d}L′=L1×L2×⋯×Ld,L′′=L1′′×L2×⋯×Ld
If L 1 L 1 ′ L_(1)^(')L_{1}^{\prime}L1′ and L 1 L 1 ′ ′ L_(1)^('')L_{1}^{\prime \prime}L1′′ (which are necessarily of the same color) intersect over a point p L 1 L 1 p ∈ L 1 ′ ∩ L 1 ′ ′ p inL_(1)^(')nnL_(1)^('')p \in L_{1}^{\prime} \cap L_{1}^{\prime \prime}p∈L1′∩L1′′ of Maslov index zero, we get a new one dimensional Lagrangian L 1 L 1 L_(1)L_{1}L1 which is a cone over p p ppp,
L 1 = Cone ( p ) = L 1 p L 1 , L 1 = Cone ⁡ ( p ) = L 1 ′ → p L 1 ′ ′ , L_(1)=Cone(p)=L_(1)^(')rarr"p"L_(1)^(''),L_{1}=\operatorname{Cone}(p)=L_{1}^{\prime} \xrightarrow{p} L_{1}^{\prime \prime},L1=Cone⁡(p)=L1′→pL1′′,
as well as a new d d ddd-dimensional Lagrangian L L LLL on Y Y YYY given by
(6.11) L = L 1 × L 2 × × L d (6.11) L = L 1 × L 2 × ⋯ × L d {:(6.11)L=L_(1)xxL_(2)xx cdots xxL_(d):}\begin{equation*} L=L_{1} \times L_{2} \times \cdots \times L_{d} \tag{6.11} \end{equation*}(6.11)L=L1×L2×⋯×Ld
The Lagrangian is a cone over the intersection point P P P\mathcal{P}P of L L ′ L^(')L^{\prime}L′ and L L ′ ′ L^('')L^{\prime \prime}L′′ which is of the form
(6.12) P = ( p , i d L 2 , , i d L d ) L L (6.12) P = p , i d L 2 , … , i d L d ∈ L ′ ∩ L ′ ′ {:(6.12)P=(p,id_(L_(2)),dots,id_(L_(d)))inL^(')nnL^(''):}\begin{equation*} \mathcal{P}=\left(p, \mathrm{id}_{L_{2}}, \ldots, \mathrm{id}_{L_{d}}\right) \in L^{\prime} \cap L^{\prime \prime} \tag{6.12} \end{equation*}(6.12)P=(p,idL2,…,idLd)∈L′∩L′′
and which also has Maslov index zero, L = Cone ( P ) L = Cone ⁡ ( P ) L=Cone(P)L=\operatorname{Cone}(\mathcal{P})L=Cone⁡(P).
Conversely, any L L LLL brane which is of the product form in (6.11) can be written as a complex [ 11 ] [ 11 ] [11][11][11]
(6.13) L L P L (6.13) L ≅ L ′ → P L ′ ′ {:(6.13)L~=L^(')rarr"P"L^(''):}\begin{equation*} L \cong L^{\prime} \xrightarrow{\mathcal{P}} L^{\prime \prime} \tag{6.13} \end{equation*}(6.13)L≅L′→PL′′
with an explicit map P P P\mathcal{P}P coming from a one-dimensional intersection point in one of its factors, as in (6.12).

6.3.5.

To find the projective resolution of the B E U B E U BEU\mathscr{B} E \mathcal{U}BEU brane in (6.5), start by isotoping the brane, by stretching it straight along the cylinder.
Let the brane break at the two infinities of A A A\mathscr{A}A, to get the direct sum brane B E ( T ) B E ( T ) BE(T)\mathscr{B} E(T)BE(T) in (6.6), on which the complex is based. To find the maps in the complex, record how the brane breaks, iterating the above construction, one one-dimensional intersection point at the time. Every intersection point of the form (6.12) translates to a specific element of the algebra A A AAA and a specific map in the complex. The result is a product of d d ddd one-dimensional complexes, which describes factors of B E U B E U BEU\mathscr{B} E \mathcal{U}BEU, and captures almost all the terms in the differential Q A Q A Q_(A)Q_{A}QA. The remaining ones follow, up to quasi-isomorphisms, by asking that the differential closes Q A 2 = 0 Q A 2 = 0 Q_(A)^(2)=0Q_{A}^{2}=0QA2=0 in the algebra A A AAA. The fact that not all terms in Q A Q A Q_(A)Q_{A}QA are geometric is a general feature of d > 1 d > 1 d > 1d>1d>1 theories.
In practice, it is convenient to first break the brane one of the two infinities of A A A\mathcal{A}A, and only then on the other. The branes at the intermediate stage are images, under the h h ∗ h^(**)h^{*}h∗ functor, of stable basis branes [ 7 , 61 ] [ 7 , 61 ] [7,61][7,61][7,61] on D X D X DX\mathscr{D} XDX. The stable basis branes play a similar role to that of Verma modules in category O O O\mathcal{O}O. The detailed algorithm is given in [5].

6.3.6.

As an example, take the left-handed trefoil and L g = s u 2 L g = s u 2 ^(L)g=su_(2){ }^{L} \mathfrak{g}=\mathfrak{s u}_{2}Lg=su2, which leads to the brane configuration from Figure 9. For simplicity, consider the reduced knot homology, where the unknot homology is set to be trivial. As in Heegard-Floer theory, this corresponds to erasing a component from the B E u B E u BEu\mathscr{B} E uBEu and the I u I u IuI uIu branes, and leads to Figure 10. This also brings us back to the setting of our running example, where Y Y YYY is the equivariant mirror to X X X\mathcal{X}X, the resolution of the A n 1 A n − 1 A_(n-1)A_{n-1}An−1 surface singularity, with n = 4 n = 4 n=4n=4n=4.

FIGURE 10

Resolution of the B E U B E U BEU\mathscr{B} E \mathcal{U}BEU brane corresponding to the reduced trefoil. The axis of the cylinder A A A\mathscr{A}A is oriented vertically here; the branes do not wind around the S 1 S 1 S^(1)S^{1}S1.
The corresponding algebra A = i , j = 0 n 1 Hom D Y ( T i , T j ) A = ⨁ i , j = 0 n − 1   Hom D Y ∗ ⁡ T i , T j A=bigoplus_(i,j=0)^(n-1)Hom_(D_(Y))^(**)(T_(i),T_(j))A=\bigoplus_{i, j=0}^{n-1} \operatorname{Hom}_{\mathscr{D}_{Y}}^{*}\left(T_{i}, T_{j}\right)A=⨁i,j=0n−1HomDY∗⁡(Ti,Tj) is the path algebra of an affine A n 1 A n − 1 A_(n-1)A_{n-1}An−1 quiver, whose nodes correspond to T i T i T_(i)T_{i}Ti branes. The arrows a i + 1 , i a i + 1 , i ∈ a_(i+1,i)ina_{i+1, i} \inai+1,i∈ Hom D Y ( T i , T i + 1 ) Hom D Y ⁡ T i , T i + 1 Hom_(D_(Y))(T_(i),T_(i+1))\operatorname{Hom}_{\mathscr{D}_{Y}}\left(T_{i}, T_{i+1}\right)HomDY⁡(Ti,Ti+1) and b i , i + 1 Hom D Y ( T i + 1 , T i { 1 } ) b i , i + 1 ∈ Hom D Y ⁡ T i + 1 , T i { 1 } b_(i,i+1)inHom_(DY)(T_(i+1),T_(i){1})b_{i, i+1} \in \operatorname{Hom}_{\mathscr{D} Y}\left(T_{i+1}, T_{i}\{1\}\right)bi,i+1∈HomDY⁡(Ti+1,Ti{1}) satisfy a i , i 1 b i 1 , i = 0 = b i , i + 1 a i + 1 , i a i , i − 1 b i − 1 , i = 0 = b i , i + 1 a i + 1 , i a_(i,i-1)b_(i-1,i)=0=b_(i,i+1)a_(i+1,i)a_{i, i-1} b_{i-1, i}=0=b_{i, i+1} a_{i+1, i}ai,i−1bi−1,i=0=bi,i+1ai+1,i, with i i iii defined modulo n n nnn. The a a aaa 's and b b bbb 's correspond to intersections of T T TTT-branes, near one or the other infinity of A A A\mathscr{A}A; we have suppressed their Λ Î› Lambda\LambdaΛ-equivariant degrees.
Isotope the B E u B E u BEu\mathscr{B} E \mathcal{u}BEu brane straight along the cylinder A A A\mathcal{A}A. Let it break into T T TTT-branes, as in Figure 10, while recording how the brane breaks, one connected sum at a time. Every connected sum of a pair of T T TTT-branes is a cone over their intersection point, at one of the two infinities of A A A\mathcal{A}A, and a specific element of the algebra A A AAA. This leads to the complex
which closes by the A A AAA-algebra relations.
The reduced homology of the trefoil is the cohomology of the complex hom A ( B E , I u { d } ) hom A ⁡ B E ∙ , I ⁡ u { d } hom_(A)(BE^(∙),Iu{d})\operatorname{hom}_{A}\left(\mathscr{B} E^{\bullet}, \operatorname{I} u\{d\}\right)homA⁡(BE∙,I⁡u{d}) in (6.9). The only non-zero contributions come from the T 2 T 2 T_(2)T_{2}T2 brane, since the cup brane I U = I 2 I U = I 2 IU=I_(2)I U=I_{2}IU=I2 is dual to it. All the maps evaluate to zero, as I U I U IUI UIU brane is a simple module for A A AAA. As a consequence,
Hom D Y ( B E u , I u [ k ] { d } ) = H k ( hom A ( B E , I 2 { d } ) ) Hom D Y ⁡ ( B E u , I u [ k ] { d } ) = H k hom A ⁡ B E ∙ , I 2 { d } Hom_(D_(Y))(BEu,Iu[k]{d})=H^(k)(hom_(A)(BE^(∙),I_(2){d}))\operatorname{Hom}_{\mathscr{D _ { Y }}}(\mathscr{B} E u, I u[k]\{d\})=H^{k}\left(\operatorname{hom}_{A}\left(\mathscr{B} E^{\bullet}, I_{2}\{d\}\right)\right)HomDY⁡(BEu,Iu[k]{d})=Hk(homA⁡(BE∙,I2{d}))
equals to Z Z Z\mathbb{Z}Z only for ( k , d ) = ( 0 , 0 ) , ( 2 , 2 ) , ( 3 , 3 ) ( k , d ) = ( 0 , 0 ) , ( 2 , − 2 ) , ( 3 , − 3 ) (k,d)=(0,0),(2,-2),(3,-3)(k, d)=(0,0),(2,-2),(3,-3)(k,d)=(0,0),(2,−2),(3,−3), and vanishes otherwise. Here, k = M k = M k=Mk=Mk=M is the Maslov or cohomological degree and d = J d = J d=Jd=Jd=J the Jones grading. This is the reduced Khovanov homology of the left-handed trefoil, up to regrading: Khovanov's ( i , j ) ( i , j ) (i,j)(i, j)(i,j) gradings are related to ( M , J ) ( M , J ) (M,J)(M, J)(M,J) by i = M + 2 J + i 0 i = M + 2 J + i 0 i=M+2J+i_(0)i=M+2 J+i_{0}i=M+2J+i0 and j = 2 J + j 0 j = 2 J + j 0 j=2J+j_(0)j=2 J+j_{0}j=2J+j0 where i 0 = 0 , j 0 = d + n + i 0 = 0 , j 0 = d + n + − i_(0)=0,j_(0)=d+n_(+)-i_{0}=0, j_{0}=d+n_{+}-i0=0,j0=d+n+− n n − n_(-)n_{-}n−, where n + = 0 , n = 3 n + = 0 , n − = 3 n_(+)=0,n_(-)=3n_{+}=0, n_{-}=3n+=0,n−=3 are the numbers of positive and negative crossings, and d = 1 d = 1 d=1d=1d=1 is the dimension of Y Y YYY [2].

6.3.7.

The theory extends to non-simply-laced Lie algebras, and to Lie superalgebras g l m n g l m ∣ n gl_(m∣n)\mathfrak{g l}_{m \mid n}glm∣n and m 2 n m ∣ 2 n _(m∣2n)\mathfrak{}_{m \mid 2 n}m∣2n, as described in [5]. The algebra A A AAA corresponding to L g L g ^(L)g{ }^{L} \mathrm{~g}L g which is a Lie superalgebra, is not an ordinary associative algebra but a differential graded algebra; the projective resolutions are then in terms of twisted complexes. This section gives a method for solving the theory which is new even for L g = g l 1 1 L g = g l 1 ∣ 1 ^(L)g=gl_(1∣1){ }^{L} \mathfrak{g}=\mathfrak{g l}_{1 \mid 1}Lg=gl1∣1, corresponding to Heegard-Floer theory. The solution differs from that in [65], in particular since our Heegard surface is A = R × S 1 A = R × S 1 A=RxxS^(1)\mathcal{A}=\mathbb{R} \times S^{1}A=R×S1, independent of the link.

ACKNOWLEDGMENTS

This work grew out of earlier collaborations with Andrei Okounkov, which were indispensable. It includes results obtained jointly with Ivan Danilenko, Elise LePage, Yixuan Li, Michael McBreen, Miroslav Rapcak, Vivek Shende, and Peng Zhou. I am grateful to all of them for collaboration. I was supported by the NSF foundation grant PHY1820912, by the Simons Investigator Award, and by the Berkeley Center for Theoretical Physics.

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MINA AGANAGIC

Department of Mathematics, University of California, Berkeley, USA and Center for Theoretical Physics, University of California, Berkeley, USA, aganagic @ berkeley.edu

VECTOR BUNDLES ON ALGEBRAIC VARIETIES

ARAVIND ASOK AND JEAN FASEL

ABSTRACT

We survey recent developments related to the problem of classifying vector bundles on algebraic varieties. We focus on the striking analogies between topology and algebraic geometry, and the way in which the Morel-Voevodsky motivic homotopy category can be used to exploit those analogies.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 14F42; Secondary 13C10, 18F25, 19A13, 55R45, 55S35

KEYWORDS

Vector bundles, motivic homotopy theory, obstruction theory

1. INTRODUCTION

The celebrated Poincaré-Hopf theorem implies that the vanishing locus of a suitably generic vector field on a closed, smooth manifold M M MMM is topologically constrained: the number of points at which a generic vector field vanishes is equal to the Euler characteristic of M M MMM. More generally, one may ask: given a vector bundle E E EEE on a compact smooth manifold, what sorts of constraints are present on the topology of vanishing loci of generic sections? If M M MMM is a connected, closed, smooth manifold of dimension d d ddd and E E EEE is a rank r r rrr vector bundle on M M MMM, then by the corank of E E EEE we will mean the difference d r d − r d-rd-rd−r. The classical work of Eilenberg, Stiefel, Steenrod, and Whitney laid down the foundations for results restricting the topology of vanishing loci of generic sections for bundles of a fixed corank; these results appear essentially in modern form in Steenrod's book [57]. For example, one knows that if the corank of E E EEE is negative, then E E EEE admits a nowhere vanishing section and if the corank of E E EEE is 0 , then a generic section vanishes at a finite set of points, and the cardinality of that finite set is determined by purely cohomological data (the Euler class of E E EEE and the corresponding Euler number of E E EEE ). The situation becomes more interesting when the corank of E E EEE is positive, to which we will return momentarily.
In the mid-1950s, Serre created a dictionary between the theory of vector bundles in topology and the theory of projective modules over a commutative ring [55, 56]. Echoing M. M. Postnikov's MathSciNet review of Serre's paper, J. F. Adams prosaically wrote in his review of H. Bass' paper [22]: "This leads to the following programme: take definitions, constructions and theorems from bundle-theory; express them as particular cases of definitions, constructions and statements about finitely-generated projective modules over a general ring; and finally, try to prove the statements under suitable assumptions". One of the results Serre presented to illustrate this dictionary was the algebro-geometric analog of existence of nowhere vanishing sections for negative corank projective modules, now frequently referred to as Serre's splitting theorem, which we recall in algebro-geometric formulation: if E E E\mathscr{E}E is a rank r r rrr vector bundle over a Noetherian affine scheme X X XXX of dimension d d ddd, then when r > d , E E O X r > d , E ≅ E ′ ⊕ O X r > d,E~=E^(')o+O_(X)r>d, \mathcal{E} \cong \mathcal{E}^{\prime} \oplus \mathcal{O}_{X}r>d,E≅E′⊕OX.
After the Pontryagin-Steenrod representability theorem, topological vector bundles on smooth manifolds (or spaces having the homotopy type of a CW complex) can be analyzed using homotopy theoretic techniques. Extending Serre's analogy further and using celebrated work of Bass, Quillen, Suslin, and Lindel, F. Morel showed that algebraic vector bundles on smooth affine varieties could be studied using an algebro-geometric homotopy theory: the Morel-Voevodsky motivic homotopy theory. In this note, we survey recent developments in the theory of algebraic vector bundles motivated by this circle of ideas, making sure to indicate the striking analogies between topology and algebraic geometry.
To give the reader a taste of the methods we will use, we mention two results here.
First, we state an improvement of Serre's splitting theorem mentioned above (for the moment it suffices to know that A 1 A 1 A^(1)\mathbb{A}^{1}A1-cohomological dimension is bounded above by Krull dimension, but can be strictly smaller). Second, we will discuss the splitting problem for projective modules in corank 1, which goes beyond any classical results.
Theorem 1.1. If k k kkk is a field, and X X XXX is a smooth affine k k kkk-scheme of A 1 A 1 A^(1)\mathbb{A}^{1}A1-cohomological dimension d ≤ d <= d\leq d≤d, then any rank r > d r > d r > dr>dr>d bundle splits off a trivial rank 1 summand.
Conjecture 1.2. Assume k k kkk is an algebraically closed field, and X = Spec R X = Spec ⁡ R X=Spec RX=\operatorname{Spec} RX=Spec⁡R is a smooth affine k k kkk-variety of dimension d d ddd. A rank d 1 d − 1 d-1d-1d−1 vector bundle & & &\mathcal{\&}& on X X XXX splits off a free rank 1 summand if and only if 0 = c d 1 ( E ) C H d 1 ( X ) 0 = c d − 1 ( E ) ∈ C H d − 1 ( X ) 0=c_(d-1)(E)inCH^(d-1)(X)0=c_{d-1}(\mathcal{E}) \in \mathrm{CH}^{d-1}(X)0=cd−1(E)∈CHd−1(X).
In Theorem 4.12 we verify Conjecture 1.2 in case d = 3 , 4 d = 3 , 4 d=3,4d=3,4d=3,4 (and k k kkk has characteristic not equal to 2). To motivate the techniques used to establish these results, we begin by analyzing topological variants of these conjectures. We close this note with a discussion of joint work with Mike Hopkins which addresses the difficult problem of constructing interesting low rank vector bundles on "simple" algebraic varieties. As with any survey, this one reflects the biases and knowledge of the authors. Limitations of space have prevented us from talking about a number of very exciting and closely related topics.

2. A FEW TOPOLOGICAL STORIES

In this section, we recall a few topological constructions that elucidate the approaches we use to analyze corresponding algebro-geometric questions studied later.

2.1. Moore-Postnikov factorizations

Suppose f : E B f : E → B f:E rarr Bf: E \rightarrow Bf:E→B is a morphism of pointed, connected topological spaces having the homotopy type of C W C W CW\mathrm{CW}CW complexes that induces an isomorphism of fundamental groups (for simplicity of discussion). Write F F FFF for the "homotopy" fiber of f f fff, so that there is a fiber sequence
F E f B F → E → f B F rarr Erarr"f"BF \rightarrow E \xrightarrow{f} BF→E→fB
yielding a long exact sequence relating the homotopy of F , E F , E F,EF, EF,E, and B B BBB.
A basic question that arises repeatedly is the following: given a map M B M → B M rarr BM \rightarrow BM→B, when can it be lifted along f f fff to a map M E M → E M rarr EM \rightarrow EM→E ? To approach this problem, one method is to factor f f fff in such a way as to break the original lifting problem into simpler problems where existence of a lift can be checked by, say, cohomological means.
One systematic approach to analyzing this question was laid out in the work of Moore-Postnikov. In this case, one factors f f fff so as to build E E EEE out of B B BBB by sequentially adding higher homotopy of f f fff (keeping track of the induced action of π 1 ( E ) π 1 ( B ) Ï€ 1 ( E ) ≅ Ï€ 1 ( B ) pi_(1)(E)~=pi_(1)(B)\pi_{1}(E) \cong \pi_{1}(B)Ï€1(E)≅π1(B) on the fiber). In more detail, the Moore-Postnikov tower of f f fff consists of a sequence of spaces τ i f , i 0 Ï„ ≤ i f , i ≥ 0 tau_( <= i)f,i >= 0\tau_{\leq i} f, i \geq 0τ≤if,i≥0 and morphisms fitting into the following diagram:
The key properties of this factorization are that (i) the composite maps E τ i f B E → Ï„ ≤ i f → B E rarrtau_( <= i)f rarr BE \rightarrow \tau_{\leq i} f \rightarrow BE→τ≤if→B all coincide with f f fff, (ii) the maps E τ i f E → Ï„ ≤ i f E rarrtau_( <= i)fE \rightarrow \tau_{\leq i} fE→τ≤if induce isomorphisms on homotopy groups in degrees i ≤ i <= i\leq i≤i, (iii) the maps τ i f B Ï„ ≤ i f → B tau_( <= i)f rarr B\tau_{\leq i} f \rightarrow Bτ≤if→B induce isomorphisms on homotopy in degrees > i + 1 > i + 1 > i+1>i+1>i+1, and (iv) there is a homotopy pullback diagram of the form
In particular, the morphism τ i f τ i 1 f Ï„ ≤ i f → Ï„ ≤ i − 1 f tau_( <= i)f rarrtau_( <= i-1)f\tau_{\leq i} f \rightarrow \tau_{\leq i-1} fτ≤if→τ≤i−1f is a twisted principal fibration, which means that a morphism M τ i 1 f M → Ï„ ≤ i − 1 f M rarrtau_( <= i-1)fM \rightarrow \tau_{\leq i-1} fM→τ≤i−1f lifts along the tower if and only if the composite M K π 1 ( E ) ( π i ( F ) , i + 1 ) M → K Ï€ 1 ( E ) Ï€ i ( F ) , i + 1 M rarrK^(pi_(1)(E))(pi_(i)(F),i+1)M \rightarrow K^{\pi_{1}(E)}\left(\pi_{i}(F), i+1\right)M→KÏ€1(E)(Ï€i(F),i+1) lifts to B π 1 ( E ) B Ï€ 1 ( E ) Bpi_(1)(E)\mathrm{B} \pi_{1}(E)BÏ€1(E). The latter map amounts to a cohomology class on M M MMM with coefficients in a local coefficient system; this cohomology class is pulled back from a "universal example" the k k kkk-invariant at the corresponding stage. If the obstruction vanishes, a lift exists. Lifts are not unique in general, but the ambiguity in choice of a lift can also be described.

2.2. The topological splitting problem

In this section, to motivate some of the algebro-geometric results we will describe later, we review the problem of deciding whether a bundle of corank 0 or 1 on a closed smooth manifold M M MMM of dimension d + 1 d + 1 d+1d+1d+1 has a nowhere vanishing section. We now phrase this problem as a lifting problem of the type described in the preceding section.
In this case, the relevant lifting problem is:
To analyze the lifting problem, we describe the Moore-Postnikov factorization of f f fff. The homotopy fiber of f f fff coincides with the standard sphere S d 1 O ( d ) / O ( d 1 ) S d − 1 ≅ O ( d ) / O ( d − 1 ) S^(d-1)~=O(d)//O(d-1)S^{d-1} \cong O(d) / O(d-1)Sd−1≅O(d)/O(d−1).
The stabilization map O ( d 1 ) O ( d ) O ( d − 1 ) → O ( d ) O(d-1)rarr O(d)O(d-1) \rightarrow O(d)O(d−1)→O(d) is compatible with the determinant, and there are thus induced isomorphisms π 1 ( B O ( d 1 ) ) π 1 ( B O ( d ) ) Z / 2 Ï€ 1 ( B O ( d − 1 ) ) → Ï€ 1 ( B O ( d ) ) ≅ Z / 2 pi_(1)(BO(d-1))rarrpi_(1)(BO(d))~=Z//2\pi_{1}(\mathrm{~B} O(d-1)) \rightarrow \pi_{1}(\mathrm{~B} O(d)) \cong \mathbb{Z} / 2Ï€1( BO(d−1))→π1( BO(d))≅Z/2 compatible with f f fff. Note, however, that the action of Z / 2 Z / 2 Z//2\mathbb{Z} / 2Z/2 on the higher homotopy of B O ( d ) B O ( d ) BO(d)\mathrm{B} O(d)BO(d) depends on the parity of d d ddd : when d d ddd is odd the action is trivial, while if d d ddd is even the action is nontrivial in general and even fails to be nilpotent. Of course, S d 1 S d − 1 S^(d-1)S^{d-1}Sd−1 is ( d 2 ) ( d − 2 ) (d-2)(d-2)(d−2)-connected.
Remark 2.1. At this stage, the fact that bundles of negative corank on spaces have the homotopy type of a CW complex of dimension d d ddd follows immediately from obstruction theory granted the assertion that the sphere S r S r S^(r)S^{r}Sr is an ( r 1 ) ( r − 1 ) (r-1)(r-1)(r−1)-connected space in conjunction with the fact that negative corank means r > d r > d r > dr>dr>d.
In order to write down obstructions, we need some information about the homotopy of spheres: the first nonvanishing homotopy group of S d 1 S d − 1 S^(d-1)S^{d-1}Sd−1 is π d 1 ( S d 1 ) Ï€ d − 1 S d − 1 pi_(d-1)(S^(d-1))\pi_{d-1}\left(S^{d-1}\right)Ï€d−1(Sd−1) which coincides
with Z Z Z\mathbb{Z}Z for all d 2 d ≥ 2 d >= 2d \geq 2d≥2 (via the degree map). Likewise, π d ( S d 1 ) Ï€ d S d − 1 pi_(d)(S^(d-1))\pi_{d}\left(S^{d-1}\right)Ï€d(Sd−1) is Z Z Z\mathbb{Z}Z if d = 3 d = 3 d=3d=3d=3 and Z / 2 Z / 2 Z//2\mathbb{Z} / 2Z/2 if d > 3 d > 3 d > 3d>3d>3 and is generated by a suitable suspension of the classical Hopf map η : S 3 S 2 η : S 3 → S 2 eta:S^(3)rarrS^(2)\eta: S^{3} \rightarrow S^{2}η:S3→S2.
Assume now X X XXX is a space having the homotopy type of a finite C W C W CW\mathrm{CW}CW complex of dimension d + 1 d + 1 d+1d+1d+1 for some fixed integer d 2 d ≥ 2 d >= 2d \geq 2d≥2 (to eliminate some uninteresting cases) and ξ : X B O ( d ) ξ : X → B O ( d ) xi:X rarrBO(d)\xi: X \rightarrow \mathrm{B} O(d)ξ:X→BO(d) classifies a rank d d ddd vector bundle on X X XXX. The first nonzero k k kkk-invariant for f f fff yields a map X K Z / 2 ( π d 1 ( S d 1 ) , d ) X → K Z / 2 Ï€ d − 1 S d − 1 , d X rarrK^(Z//2)(pi_(d-1)(S^(d-1)),d)X \rightarrow K^{\mathbb{Z} / 2}\left(\pi_{d-1}\left(S^{d-1}\right), d\right)X→KZ/2(Ï€d−1(Sd−1),d), i.e., an element
e ( ξ ) H d ( X , Z [ σ ] ) e ( ξ ) ∈ H d ( X , Z [ σ ] ) e(xi)inH^(d)(X,Z[sigma])e(\xi) \in H^{d}(X, \mathbb{Z}[\sigma])e(ξ)∈Hd(X,Z[σ])
called the (twisted) Euler class, where Z [ σ ] Z [ σ ] Z[sigma]\mathbb{Z}[\sigma]Z[σ] is Z Z Z\mathbb{Z}Z twisted by the orientation character σ σ sigma\sigmaσ defined by applying π 1 Ï€ 1 pi_(1)\pi_{1}Ï€1 to the morphism X B O ( d ) B ( Z / 2 ) X → B O ( d ) → B ( Z / 2 ) X rarrBO(d)rarrB(Z//2)X \rightarrow \mathrm{B} O(d) \rightarrow \mathrm{B}(\mathbb{Z} / 2)X→BO(d)→B(Z/2).
Assuming this primary obstruction vanishes, one may choose a lift to the next stage of the Postnikov tower. If we fix a lift, then there is a well-defined secondary obstruction to lifting, that comes from the next k k kkk-invariant: this obstruction is given by a map X K Z / 2 ( π d ( S d 1 ) , d + 1 ) X → K Z / 2 Ï€ d S d − 1 , d + 1 X rarrK^(Z//2)(pi_(d)(S^(d-1)),d+1)X \rightarrow K^{\mathbb{Z} / 2}\left(\pi_{d}\left(S^{d-1}\right), d+1\right)X→KZ/2(Ï€d(Sd−1),d+1), i.e., a cohomology class in H d + 1 ( X , Z [ σ ] ) H d + 1 ( X , Z [ σ ] ) H^(d+1)(X,Z[sigma])H^{d+1}(X, \mathbb{Z}[\sigma])Hd+1(X,Z[σ]) if d = 3 d = 3 d=3d=3d=3 or H d + 1 ( X , Z / 2 ) H d + 1 ( X , Z / 2 ) H^(d+1)(X,Z//2)H^{d+1}(X, \mathbb{Z} / 2)Hd+1(X,Z/2) if d 3 d ≠ 3 d!=3d \neq 3d≠3; in the latter case the choice of orientation character no longer affects this cohomology group.
If one tracks the effect of choice of lift on the obstruction class described above, one obtains a map K Z / 2 ( π d 1 ( S d 1 ) , d 1 ) K Z / 2 ( π d ( S d 1 ) , d + 1 ) K Z / 2 Ï€ d − 1 S d − 1 , d − 1 → K Z / 2 Ï€ d S d − 1 , d + 1 K^(Z//2)(pi_(d-1)(S^(d-1)),d-1)rarrK^(Z//2)(pi_(d)(S^(d-1)),d+1)K^{\mathbb{Z} / 2}\left(\pi_{d-1}\left(S^{d-1}\right), d-1\right) \rightarrow K^{\mathbb{Z} / 2}\left(\pi_{d}\left(S^{d-1}\right), d+1\right)KZ/2(Ï€d−1(Sd−1),d−1)→KZ/2(Ï€d(Sd−1),d+1), which is a twisted cohomology operation. If d = 3 d = 3 d=3d=3d=3, the map in question is a twisted version of the Pontryagin squaring operation, while if d > 3 d > 3 d > 3d>3d>3 the operation can be described as S q 2 + w 2 S q 2 + w 2 ∪ Sq^(2)+w_(2)uu\mathrm{Sq}^{2}+w_{2} \cupSq2+w2∪, where w 2 w 2 w_(2)w_{2}w2 is the second Stiefel-Whitney class of the bundle. In that case, the secondary obstruction yields a well-defined coset in
o 2 ( ξ ) H d + 1 ( X , Z / 2 ) / ( S q 2 + w 2 ) H d 1 ( X , Z [ σ ] ) o 2 ( ξ ) ∈ H d + 1 ( X , Z / 2 ) / S q 2 + w 2 ∪ H d − 1 ( X , Z [ σ ] ) o_(2)(xi)inH^(d+1)(X,Z//2)//(Sq^(2)+w_(2)uu)H^(d-1)(X,Z[sigma])o_{2}(\xi) \in H^{d+1}(X, \mathbb{Z} / 2) /\left(\mathrm{Sq}^{2}+w_{2} \cup\right) H^{d-1}(X, \mathbb{Z}[\sigma])o2(ξ)∈Hd+1(X,Z/2)/(Sq2+w2∪)Hd−1(X,Z[σ])
This description of the primary and secondary obstructions was laid out carefully by the early 1950s by S. D. Liao [37].
Finally, the dimension assumption on X X XXX guarantees that a lift of ξ ξ xi\xiξ along f f fff exists if and only if these two obstructions vanish. In principle, this kind of analysis can be continued, though the calculations become more involved as the indeterminacy created by successive choices of lifts becomes harder to control and information about higher unstable homotopy of spheres is also harder to obtain. For a thorough treatment of this and even more general situations, we refer the reader to [61].
Remark 2.2. The analysis of the obstructions can be improved by organizing the calculations differently. The Moore-Postnikov factorization has the effect of factoring a map f : X Y f : X → Y f:X rarr Yf: X \rightarrow Yf:X→Y as a tower of fibrations where the relevant fibers are Eilenberg-Mac Lane spaces. However, there are many other ways to produce factorizations of f f fff with different constraints on the "cohomological" properties of pieces of the tower.

3. A QUICK REVIEW OF MOTIVIC HOMOTOPY THEORY

Motivic homotopy theory, introduced by F. Morel and V. Voevodsky [41], provides a homotopy theory for schemes over a base. While there are a number of different approaches
to constructing the motivic homotopy category that work in great generality, we work in a very concrete situation. By an algebraic variety over a field k k kkk, we will mean a separated, finite type, reduced k k kkk-scheme. We write S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk for the category of smooth algebraic varieties; for later use, we will also write S m k aff S m k aff  Sm_(k)^("aff ")\mathrm{Sm}_{k}^{\text {aff }}Smkaff  for the full subcategory of S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk consisting of affine schemes.
The category S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk is "too small" to do homotopy theory, in the sense that various natural categorical constructions one would like to make (increasing unions, quotients by subspaces, etc.) can leave the category. As such, one first enlarges S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk to a suitable category S p c k S p c k Spc_(k)\mathrm{Spc}_{k}Spck of "spaces"; one may take S p c k S p c k Spc_(k)\mathrm{Spc}_{k}Spck to be the category of simplicial presheaves on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk and the functor S m k S p c k S m k → S p c k Sm_(k)rarrSpc_(k)\mathrm{Sm}_{k} \rightarrow \mathrm{Spc}_{k}Smk→Spck is given by the Yoneda embedding followed by the functor viewing a presheaf on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk as a constant simplicial presheaf.

3.1. Homotopical sheaf theory

Passing to S p c k S p c k Spc_(k)\mathrm{Spc}_{k}Spck has the effect of destroying certain colimits that one would like to retain. To recover the colimits that have been lost, one localizes S p c k S p c k Spc_(k)\mathrm{Spc}_{k}Spck and passes to a suitable "local" homotopy category of the sort first studied in detail by K. Brown-S. Gersten, A. Joyal, and J.F. Jardine: one fixes a Grothendieck topology τ Ï„ tau\tauÏ„ on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk and inverts the so-called τ Ï„ tau\tauÏ„-local weak equivalences on S p c k S p c k Spc_(k)\mathrm{Spc}_{k}Spck; we refer the reader to [34] for a textbook treatment. We write H τ ( k ) H Ï„ ( k ) H_(tau)(k)\mathrm{H}_{\tau}(k)HÏ„(k) for the resulting localization of S p c k S p c k Spc_(k)\mathrm{Spc}_{k}Spck. If X S p c k X ∈ S p c k XinSpc_(k)\mathscr{X} \in \mathrm{Spc}_{k}X∈Spck, then a base-point for X X X\mathscr{X}X is a morphism x : Spec k X x : Spec ⁡ k → X x:Spec k rarrXx: \operatorname{Spec} k \rightarrow \mathscr{X}x:Spec⁡k→X splitting the structure morphism. There is an associated pointed homotopy category and these homotopy categories can be thought of as providing a convenient framework for "nonabelian" homological algebra.
Henceforth, we take τ Ï„ tau\tauÏ„ to be the Nisnevich topology (which is finer than the Zariski topology, but coarser than the étale topology). For the purposes of this note, it suffices to observe that the Nisnevich cohomological dimension of a k k kkk-scheme is equal to its Krull dimension, like the Zariski topology.
In the category of pointed spaces, we can make sense of wedge sums and smash products, just as in ordinary topology. We also define spheres S i , i 0 S i , i ≥ 0 S^(i),i >= 0S^{i}, i \geq 0Si,i≥0, as the constant simplicial presheaves corresponding to the simplicial sets S i S i S^(i)S^{i}Si. For any pointed space ( X , x ) ( X , x ) (X,x)(\mathscr{X}, x)(X,x), we define its homotopy sheaves π i ( X , x ) Ï€ i ( X , x ) pi_(i)(X,x)\pi_{i}(\mathscr{X}, x)Ï€i(X,x) as the Nisnevich sheaves associated with the presheaves on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk defined by
U hom H Nis ( k ) ( S i U + , X , x ) U ↦ hom H Nis  ( k ) ⁡ S i ∧ U + , X , x U|->hom_(H_("Nis ")(k))(S^(i)^^U_(+),X,x)U \mapsto \operatorname{hom}_{\mathrm{H}_{\text {Nis }}(k)}\left(S^{i} \wedge U_{+}, \mathscr{X}, x\right)U↦homHNis (k)⁡(Si∧U+,X,x)
here the subscript + means adjoint a disjoint base-point. These homotopy sheaves may be used to formulate a Whitehead theorem.
If G G G\mathbf{G}G is a Nisnevich sheaf of groups on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk, then there is a classifying space B G such that for any smooth k k kkk-scheme X X XXX one has a functorial identification of pointed sets of the form
hom H N i s ( k ) ( X , B G ) = H N i s 1 ( X , G ) hom H N i s ( k ) ⁡ ( X , B G ) = H N i s 1 ( X , G ) hom_(H_(Nis)(k))(X,BG)=H_(Nis)^(1)(X,G)\operatorname{hom}_{\mathrm{H}_{\mathrm{Nis}}(k)}(X, \mathrm{~B} \mathbf{G})=\mathrm{H}_{\mathrm{Nis}}^{1}(X, \mathbf{G})homHNis(k)⁡(X, BG)=HNis1(X,G)
For later use, we set
Vect n ( X ) := H Z a r 1 ( X , G L n ) = H N i s 1 ( X , G L n ) = hom H N i s ( k ) ( X , B G L n ) Vect n ⁡ ( X ) := H Z a r 1 X , G L n = H N i s 1 X , G L n = hom H N i s ( k ) ⁡ X , B G L n Vect_(n)(X):=H_(Zar)^(1)(X,GL_(n))=H_(Nis)^(1)(X,GL_(n))=hom_(H_(Nis)(k))(X,BGL_(n))\operatorname{Vect}_{n}(X):=\mathrm{H}_{\mathrm{Zar}}^{1}\left(X, \mathrm{GL}_{n}\right)=\mathrm{H}_{\mathrm{Nis}}^{1}\left(X, \mathrm{GL}_{n}\right)=\operatorname{hom}_{\mathrm{H}_{\mathrm{Nis}}(k)}\left(X, \mathrm{BGL}_{n}\right)Vectn⁡(X):=HZar1(X,GLn)=HNis1(X,GLn)=homHNis(k)⁡(X,BGLn)
where we as usual identify isomorphism classes of rank n rank ⁡ n rank n\operatorname{rank} nrank⁡n vector bundles locally trivial with respect to the Zariski topology on X X XXX with G L n G L n GL_(n)\mathrm{GL}_{n}GLn-torsors (and the choice of topology does not matter).
If A A A\mathbf{A}A is any Nisnevich sheaf of abelian groups on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk, then for any integer n 0 n ≥ 0 n >= 0n \geq 0n≥0 there are Eilenberg-Mac Lane spaces K ( A , n ) K ( A , n ) K(A,n)\mathrm{K}(\mathbf{A}, n)K(A,n), i.e., spaces with exactly one nonvanishing homotopy sheaf, appearing in degree n n nnn, isomorphic to A A A\mathbf{A}A. For such spaces, hom H Nis ( k ) ( X , K ( A , n ) ) hom H Nis  ( k ) ⁡ ( X , K ( A , n ) ) hom_(H_("Nis ")(k))(X,K(A,n))\operatorname{hom}_{\mathrm{H}_{\text {Nis }}(k)}(X, \mathrm{~K}(\mathbf{A}, n))homHNis (k)⁡(X, K(A,n)) has a natural abelian group structure, and there are functorial isomorphisms of abelian groups
hom H N i s ( k ) ( X , K ( A , n ) ) = H N i s n ( X , A ) hom H N i s ( k ) ⁡ ( X , K ( A , n ) ) = H N i s n ( X , A ) hom_(H_(Nis)(k))(X,K(A,n))=H_(Nis)^(n)(X,A)\operatorname{hom}_{\mathrm{H}_{\mathrm{Nis}}(k)}(X, \mathrm{~K}(\mathbf{A}, n))=\mathrm{H}_{\mathrm{Nis}}^{n}(X, \mathbf{A})homHNis(k)⁡(X, K(A,n))=HNisn(X,A)
With this definition, for essentially formal reasons there is a suspension isomorphism for Nisnevich cohomology with respect to the suspension S 1 ( ) S 1 ∧ ( − ) S^(1)^^(-)S^{1} \wedge(-)S1∧(−).

3.2. The motivic homotopy category

The motivic homotopy category is obtained as a further localization of H N i s ( k ) H N i s ( k ) H_(Nis)(k)\mathrm{H}_{\mathrm{Nis}}(k)HNis(k) : one localizes at the projection morphisms X × A 1 X X × A 1 → X XxxA^(1)rarrX\mathscr{X} \times \mathbb{A}^{1} \rightarrow \mathscr{X}X×A1→X. We write H ( k ) H ( k ) H(k)\mathrm{H}(k)H(k) for the resulting homotopy category; isomorphisms in this category will be referred to as A 1 A 1 A^(1)\mathbb{A}^{1}A1-weak equivalences. Following the notation in classical homotopy theory, we write
[ X , Y ] A 1 := hom H ( k ) ( X , Y ) [ X , Y ] A 1 := hom H ( k ) ⁡ ( X , Y ) [X,Y]_(A^(1)):=hom_(H(k))(X,Y)[\mathscr{X}, \mathscr{Y}]_{\mathbb{A}^{1}}:=\operatorname{hom}_{\mathrm{H}(k)}(\mathscr{X}, \mathscr{Y})[X,Y]A1:=homH(k)⁡(X,Y)
and refer to this set as the set of A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy classes of maps from X X X\mathscr{X}X to Y Y Y\mathscr{Y}Y.
If X X X\mathscr{X}X is a space, we will write π 0 A 1 ( X ) Ï€ 0 A 1 ( X ) pi_(0)^(A^(1))(X)\pi_{0}^{\mathbb{A}^{1}}(\mathscr{X})Ï€0A1(X) for the Nisnevich sheaf associated with the presheaf U [ U , X ] A 1 U ↦ [ U , X ] A 1 U|->[U,X]_(A^(1))U \mapsto[U, \mathscr{X}]_{\mathbb{A}^{1}}U↦[U,X]A1 on Sm k Sm k Sm_(k)\operatorname{Sm}_{k}Smk; we refer to π 0 A 1 ( X ) Ï€ 0 A 1 ( X ) pi_(0)^(A^(1))(X)\pi_{0}^{\mathbb{A}^{1}}(\mathscr{X})Ï€0A1(X) as the sheaf of connected components, and we say that X X X\mathscr{X}X is A 1 A 1 A^(1)\mathbb{A}^{1}A1-connected if π 0 A 1 ( X ) Ï€ 0 A 1 ( X ) pi_(0)^(A^(1))(X)\pi_{0}^{\mathbb{A}^{1}}(\mathscr{X})Ï€0A1(X) is the sheaf Spec ( k ) Spec ⁡ ( k ) Spec(k)\operatorname{Spec}(k)Spec⁡(k).
We consider G m G m G_(m)\mathbb{G}_{m}Gm as a pointed space, with base point its identity section 1 . In that case, we define motivic spheres
S i , j := S i G m j S i , j := S i ∧ G m ∧ j S^(i,j):=S^(i)^^G_(m)^(^^j)S^{i, j}:=S^{i} \wedge \mathbb{G}_{m}^{\wedge j}Si,j:=Si∧Gm∧j
We caution the reader that there are a number of different indexing conventions used for motivic spheres. One defines bigraded homotopy sheaves π i , j A 1 ( X , x ) Ï€ i , j A 1 ( X , x ) pi_(i,j)^(A^(1))(X,x)\pi_{i, j}^{\mathbb{A}^{1}}(\mathscr{X}, x)Ï€i,jA1(X,x) for any pointed space as the Nisnevich sheaves associated with the presheaves on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk
U [ S i , j U + , X ] A 1 U ↦ S i , j ∧ U + , X A 1 U|->[S^(i,j)^^U_(+),X]_(A^(1))U \mapsto\left[S^{i, j} \wedge U_{+}, \mathscr{X}\right]_{\mathbb{A}^{1}}U↦[Si,j∧U+,X]A1
we write π i A 1 ( X , x ) Ï€ i A 1 ( X , x ) pi_(i)^(A^(1))(X,x)\pi_{i}^{\mathbb{A}^{1}}(\mathscr{X}, x)Ï€iA1(X,x) for π i , 0 A 1 ( X ) Ï€ i , 0 A 1 ( X ) pi_(i,0)^(A^(1))(X)\pi_{i, 0}^{\mathbb{A}^{1}}(\mathscr{X})Ï€i,0A1(X). We will say that a pointed space ( X , x ) ( X , x ) (X,x)(\mathscr{X}, x)(X,x) is A 1 A 1 A^(1)\mathbb{A}^{1}A1 - k k kkk-connected for some integer k 1 k ≥ 1 k >= 1k \geq 1k≥1 if it is A 1 A 1 A^(1)\mathbb{A}^{1}A1-connected and the sheaves π i A 1 ( X , x ) Ï€ i A 1 ( X , x ) pi_(i)^(A^(1))(X,x)\pi_{i}^{\mathbb{A}^{1}}(\mathscr{X}, x)Ï€iA1(X,x) are trivial for 1 i k 1 ≤ i ≤ k 1 <= i <= k1 \leq i \leq k1≤i≤k. Because of the form of the Whitehead theorem in the Nisnevich local homotopy category, the sheaves π i A 1 ( ) Ï€ i A 1 ( − ) pi_(i)^(A^(1))(-)\pi_{i}^{\mathbb{A}^{1}}(-)Ï€iA1(−) detect A 1 A 1 A^(1)\mathbb{A}^{1}A1-weak equivalences.
We write Δ k Δ k ∙ Delta_(k)^(∙)\Delta_{k}^{\bullet}Δk∙ for the cosimplicial affine space with
Δ k n := Spec k [ x 0 , , x n ] / i x i = 1 Δ k n := Spec ⁡ k x 0 , … , x n / ∑ i   x i = 1 Delta_(k)^(n):=Spec k[x_(0),dots,x_(n)]//(:sum_(i)x_(i)=1:)\Delta_{k}^{n}:=\operatorname{Spec} k\left[x_{0}, \ldots, x_{n}\right] /\left\langle\sum_{i} x_{i}=1\right\rangleΔkn:=Spec⁡k[x0,…,xn]/⟨∑ixi=1⟩
For any space X X X\mathscr{X}X, we write Sing A 1 X Sing A 1 ⁡ X Sing^(A^(1))X\operatorname{Sing}^{\mathbb{A}^{1}} \mathscr{X}SingA1⁡X for the space diag h o m _ _ ( Δ , X ) diag ⁡ h o m _ _ Δ ∙ , X diag hom__(Delta^(∙),X)\operatorname{diag} \underline{\underline{h o m}}\left(\Delta^{\bullet}, \mathscr{X}\right)diag⁡hom__(Δ∙,X). There is a canonical map X Sing A 1 X X → Sing A 1 ⁡ X XrarrSing^(A^(1))X\mathscr{X} \rightarrow \operatorname{Sing}^{\mathbb{A}^{1}} \mathscr{X}X→SingA1⁡X and the space Sing A 1 X Sing A 1 ⁡ X Sing^(A^(1))X\operatorname{Sing}^{\mathbb{A}^{1}} \mathscr{X}SingA1⁡X is called the singular construction on X X X\mathscr{X}X. For
a smooth scheme U U UUU, the set of connected components π 0 ( Sing A 1 X ( U ) ) Ï€ 0 Sing A 1 ⁡ X ( U ) pi_(0)(Sing^(A^(1))X(U))\pi_{0}\left(\operatorname{Sing}^{\mathbb{A}^{1}} \mathscr{X}(U)\right)Ï€0(SingA1⁡X(U)) will be called the set of naive A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy classes of maps U X U → X U rarrXU \rightarrow \mathscr{X}U→X (by construction, it is the quotient of the set of morphisms U X U → X U rarrXU \rightarrow \mathscr{X}U→X by the equivalence relation generated by maps U × A 1 X U × A 1 → X U xxA^(1)rarrXU \times \mathbb{A}^{1} \rightarrow \mathscr{X}U×A1→X ). Again, by definition there is a comparison morphism
(3.1) π 0 ( Sing A 1 X ( U ) ) [ U , X ] A 1 (3.1) Ï€ 0  Sing  A 1 X ( U ) → [ U , X ] A 1 {:(3.1)pi_(0)(" Sing "^(A^(1))X(U))rarr[U","X]_(A^(1)):}\begin{equation*} \pi_{0}\left(\text { Sing }^{\mathbb{A}^{1}} \mathscr{X}(U)\right) \rightarrow[U, \mathscr{X}]_{\mathbb{A}^{1}} \tag{3.1} \end{equation*}(3.1)Ï€0( Sing A1X(U))→[U,X]A1
Typically, the map (3.1) is far from being a bijection.

3.3. A 1 A 1 A^(1)\mathbb{A}^{1}A1-weak equivalences

We now give a number of examples of A 1 A 1 A^(1)\mathbb{A}^{1}A1-weak equivalences, highlighting some examples and constructions that will be important in the sequel.
Example 3.1. A smooth k k kkk-scheme X X XXX is called A 1 A 1 A^(1)\mathbb{A}^{1}A1-contractible if the structure morphism X Spec k X → Spec ⁡ k X rarr Spec kX \rightarrow \operatorname{Spec} kX→Spec⁡k is an A 1 A 1 A^(1)\mathbb{A}^{1}A1-weak equivalence. By construction, A n A n A^(n)\mathbb{A}^{n}An is an A 1 A 1 A^(1)\mathbb{A}^{1}A1-contractible smooth k k kkk-scheme. However, there are a plethora of A 1 A 1 A^(1)\mathbb{A}^{1}A1-contractible smooth k k kkk-schemes that are nonisomorphic to A n A n A^(n)\mathbb{A}^{n}An. For instance, the Russell cubic threefold, defined by the hypersurface equation x + x 2 y + z 2 + t 3 = 0 x + x 2 y + z 2 + t 3 = 0 x+x^(2)y+z^(2)+t^(3)=0x+x^{2} y+z^{2}+t^{3}=0x+x2y+z2+t3=0 is known to be nonisomorphic to affine space and also A 1 A 1 A^(1)\mathbb{A}^{1}A1-contractible [28]. See [16] for a survey of further examples.
Example 3.2. If f : X Y f : X → Y f:X rarr Yf: X \rightarrow Yf:X→Y is a Nisnevich locally trivial morphism with fibers that are A 1 A 1 A^(1)\mathbb{A}^{1}A1-contractible smooth k k kkk-schemes, then f f fff is an A 1 A 1 A^(1)\mathbb{A}^{1}A1-weak equivalence. Thus, the projection morphism for a vector bundle is an A 1 A 1 A^(1)\mathbb{A}^{1}A1-weak equivalence. A vector bundle E E EEE over a scheme X X XXX can be seen as a commutative algebraic X X XXX-group scheme, so we may speak of E E EEE-torsors; E E EEE-torsors are classified by the coherent cohomology group H 1 ( X , E ) H 1 ( X , E ) H^(1)(X,E)\mathrm{H}^{1}(X, \mathscr{E})H1(X,E) (in particular, vector bundle torsors over affine schemes may always be trivialized). Vector bundle torsors are Zariski locally trivial fiber bundles with fibers isomorphic to affine spaces, and the projection morphism for a vector bundle torsor is an A 1 A 1 A^(1)\mathbb{A}^{1}A1-weak equivalence.
By an affine vector bundle torsor over a scheme X X XXX we will mean a torsor π : Y X Ï€ : Y → X pi:Y rarr X\pi: Y \rightarrow XÏ€:Y→X for some vector bundle E E EEE on X X XXX such that Y Y YYY is an affine scheme. Jouanolou proved [35, LEMME 1.5] that any quasiprojective variety admits an affine vector bundle torsor. Thomason [63, PROPOSITION 4.4] generalized Jouanolou's observation, and the following result is a special case of his results.
Lemma 3.3 (Jouanolou-Thomason homotopy lemma). If X X XXX is a smooth k k kkk-variety, then X X XXX admits an affine vector bundle torsor. In particular, any smooth k k kkk-variety is isomorphic in H ( k ) H ( k ) H(k)\mathrm{H}(k)H(k) to a smooth affine variety.
Definition 3.4. By a Jouanolou device for a smooth k k kkk-variety X X XXX we will mean a choice of an affine vector bundle torsor p : Y X p : Y → X p:Y rarr Xp: Y \rightarrow Xp:Y→X.
Example 3.5. When X = P n X = P n X=P^(n)X=\mathbb{P}^{n}X=Pn there is a very simple construction of a "standard" Jouanolou device P ~ n P ~ n tilde(P)^(n)\tilde{\mathbb{P}}^{n}P~n. Geometrically, the standard Jouanolou device for P n P n P^(n)\mathbb{P}^{n}Pn may be described as the complement of the incidence divisor in P n × P n P n × P n P^(n)xxP^(n)\mathbb{P}^{n} \times \mathbb{P}^{n}Pn×Pn where the second projective space is viewed as the dual of the first, with structure morphism the projection onto either factor.
Example 3.6. If X X XXX is a smooth projective variety of dimension d d ddd, then we may choose a finite morphism ψ : X P d ψ : X → P d psi:X rarrP^(d)\psi: X \rightarrow \mathbb{P}^{d}ψ:X→Pd. Pulling back the standard Jouanolou device for P d P d P^(d)\mathbb{P}^{d}Pd along ψ ψ psi\psiψ, we see that X X XXX admits a Jouanolou device X ~ X ~ tilde(X)\tilde{X}X~ of dimension 2 d 2 d 2d2 d2d.
Example 3.7. For n N n ∈ N n inNn \in \mathbb{N}n∈N, consider the smooth affine k k kkk-scheme Q 2 n 1 Q 2 n − 1 Q_(2n-1)Q_{2 n-1}Q2n−1 defined as the hypersurface in A k 2 n A k 2 n A_(k)^(2n)\mathbb{A}_{k}^{2 n}Ak2n given by the equation i = 1 n x i y i = 1 ∑ i = 1 n   x i y i = 1 sum_(i=1)^(n)x_(i)y_(i)=1\sum_{i=1}^{n} x_{i} y_{i}=1∑i=1nxiyi=1. Projecting onto the first n n nnn-factors, we obtain a map p : Q 2 n 1 A n 0 p : Q 2 n − 1 → A n ∖ 0 p:Q_(2n-1)rarrA^(n)\\0p: Q_{2 n-1} \rightarrow \mathbb{A}^{n} \backslash 0p:Q2n−1→An∖0 which one may check is an affine vector bundle torsor. For any integer n 0 , A n 0 n ≥ 0 , A n ∖ 0 n >= 0,A^(n)\\0n \geq 0, \mathbb{A}^{n} \backslash 0n≥0,An∖0 is A 1 A 1 A^(1)\mathbb{A}^{1}A1-weakly equivalent to S n 1 , n S n − 1 , n S^(n-1,n)S^{n-1, n}Sn−1,n (see [41, §3.2, EXAMPLE 2.20]) and consequently Q 2 n 1 Q 2 n − 1 Q_(2n-1)Q_{2 n-1}Q2n−1 is A 1 A 1 A^(1)\mathbb{A}^{1}A1-weakly equivalent to S n 1 , n S n − 1 , n S^(n-1,n)S^{n-1, n}Sn−1,n as well.
Example 3.8. For n N n ∈ N n inNn \in \mathbb{N}n∈N, consider the smooth affine k k kkk-scheme Q 2 n Q 2 n Q_(2n)Q_{2 n}Q2n defined as the hypersurface in A k 2 n + 1 A k 2 n + 1 A_(k)^(2n+1)\mathbb{A}_{k}^{2 n+1}Ak2n+1 given by the equation
i = 1 n x i y i = z ( 1 z ) ∑ i = 1 n   x i y i = z ( 1 − z ) sum_(i=1)^(n)x_(i)y_(i)=z(1-z)\sum_{i=1}^{n} x_{i} y_{i}=z(1-z)∑i=1nxiyi=z(1−z)
The variety Q 2 Q 2 Q_(2)Q_{2}Q2 is isomorphic to the standard Jouanolou device over P 1 P 1 P^(1)\mathbb{P}^{1}P1. The variety P 1 P 1 P^(1)\mathbb{P}^{1}P1 is A 1 A 1 A^(1)\mathbb{A}^{1}A1-weakly equivalent to S 1 , 1 S 1 , 1 S^(1,1)S^{1,1}S1,1 and therefore Q 2 Q 2 Q_(2)Q_{2}Q2 is A 1 A 1 A^(1)\mathbb{A}^{1}A1-weakly equivalent to S 1 , 1 S 1 , 1 S^(1,1)S^{1,1}S1,1 as well. For n 2 n ≥ 2 n >= 2n \geq 2n≥2, one knows that Q 2 n Q 2 n Q_(2n)Q_{2 n}Q2n is A 1 A 1 A^(1)\mathbb{A}^{1}A1-weakly equivalent to S n , n S n , n S^(n,n)S^{n, n}Sn,n [2, THEOREM 2].

3.4. Representability results

If F F F\mathscr{F}F is a presheaf on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk, we will say that F F F\mathscr{F}F is A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant (resp. A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant on affines) if the pullback map F ( X ) F ( X × A 1 ) F ( X ) → F X × A 1 F(X)rarrF(X xxA^(1))\mathscr{F}(X) \rightarrow \mathscr{F}\left(X \times \mathbb{A}^{1}\right)F(X)→F(X×A1) is an isomorphism for all X Sm k X ∈ Sm k X inSm_(k)X \in \operatorname{Sm}_{k}X∈Smk (resp. X S m k aff X ∈ S m k aff  X inSm_(k)^("aff ")X \in \mathrm{Sm}_{k}^{\text {aff }}X∈Smkaff  ). A necessary condition for a cohomology theory on smooth schemes to be representable in H ( k ) H ( k ) H(k)\mathrm{H}(k)H(k) is that it is A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant and has a Mayer-Vietoris property with respect to the Nisnevich topology. One of the first functors that one encounters with these properties is that which assigns to a smooth k k kkk-scheme its Picard group. Morel and Voevodsky showed [41, $4 PROPOSITION 3.8] that if X X XXX is a smooth k k kkk-scheme, then the A 1 A 1 A^(1)\mathbb{A}^{1}A1-weak equivalence P B G m P ∞ → B G m P^(oo)rarrBG_(m)\mathbb{P}^{\infty} \rightarrow \mathrm{B} \mathbb{G}_{m}P∞→BGm induces a bijection [ X , P ] A 1 Pic ( X ) X , P ∞ A 1 ≅ Pic ⁡ ( X ) [X,P^(oo)]_(A^(1))~=Pic(X)\left[X, \mathbb{P}^{\infty}\right]_{\mathbb{A}^{1}} \cong \operatorname{Pic}(X)[X,P∞]A1≅Pic⁡(X).
If A A A\mathbf{A}A is a sheaf of abelian groups on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk, then the functors H N i s i ( , A ) H N i s i ( − , A ) H_(Nis)^(i)(-,A)\mathrm{H}_{\mathrm{Nis}}^{i}(-, \mathbf{A})HNisi(−,A) frequently fail to be A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant (taking A = G a A = G a A=G_(a)\mathbf{A}=\mathbb{G}_{a}A=Ga gives a simple example) and therefore fail to be representable on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk. The situation above where A = G m A = G m A=G_(m)\mathbf{A}=\mathbb{G}_{m}A=Gm provides the prototypical example of a sheaf whose cohomology is A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant (here the zeroth cohomology is the presheaf of units, which is even A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant on reduced schemes). Following Morel and Voevodsky, we distinguish the cases where sheaf cohomology is A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant.
Definition 3.9. A sheaf of groups G G G\mathbf{G}G on Sm k Sm k Sm_(k)\operatorname{Sm}_{k}Smk is called strongly A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant if for i = 0 , 1 i = 0 , 1 i=0,1i=0,1i=0,1 the functors H N i s i ( , G ) H N i s i ( − , G ) H_(Nis)^(i)(-,G)\mathrm{H}_{\mathrm{Nis}}^{i}(-, \mathbf{G})HNisi(−,G) on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk are A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant. A sheaf of abelian groups A A A\mathbf{A}A on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk is called strictly A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant if for all i 0 i ≥ 0 i >= 0i \geq 0i≥0 the functors H N i s i ( , A ) H N i s i ( − , A ) H_(Nis)^(i)(-,A)\mathrm{H}_{\mathrm{Nis}}^{i}(-, \mathbf{A})HNisi(−,A) on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk are A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant.
The fundamental work of Morel, which we will review shortly, demonstrates the key role played by strongly and strictly A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant sheaves. Nevertheless, various natural functors of geometric origin fail to be A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant on smooth schemes.
Example 3.10. If r 2 r ≥ 2 r >= 2r \geq 2r≥2, then the functor H N i s 1 ( , G L r ) H N i s 1 − , G L r H_(Nis)^(1)(-,GL_(r))\mathrm{H}_{\mathrm{Nis}}^{1}\left(-, \mathrm{GL}_{r}\right)HNis1(−,GLr) fails to be A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant on all schemes. For an explicit example, consider the simplest case. By a theorem of DedekindWeber frequently attributed to Grothendieck every rank n n nnn vector bundle on P 1 P 1 P^(1)\mathbb{P}^{1}P1 is isomorphic to a unique line bundle of the form i = 1 n O ( a i ) ⨁ i = 1 n   O a i bigoplus_(i=1)^(n)O(a_(i))\bigoplus_{i=1}^{n} \mathcal{O}\left(a_{i}\right)⨁i=1nO(ai) with the a i a i a_(i)a_{i}ai weakly increasing. On the other hand, consider P 1 × A 1 P 1 × A 1 P^(1)xxA^(1)\mathbb{P}^{1} \times \mathbb{A}^{1}P1×A1 with coordinates t t ttt and x x xxx. The matrix
( t x 0 t 1 ) t x 0 t − 1 ([t,x],[0,t^(-1)])\left(\begin{array}{cc} t & x \\ 0 & t^{-1} \end{array}\right)(tx0t−1)
determines a rank 2 vector bundle on P 1 × A 1 P 1 × A 1 P^(1)xxA^(1)\mathbb{P}^{1} \times \mathbb{A}^{1}P1×A1 whose restriction to P 1 × 0 P 1 × 0 P^(1)xx0\mathbb{P}^{1} \times 0P1×0 is O ( 1 ) O ( 1 ) O ( 1 ) ⊕ O ( − 1 ) O(1)o+O(-1)\mathcal{O}(1) \oplus \mathcal{O}(-1)O(1)⊕O(−1) and whose restriction to P 1 × 1 P 1 × 1 P^(1)xx1\mathbb{P}^{1} \times 1P1×1 is O O O ⊕ O Oo+O\mathcal{O} \oplus \mathcal{O}O⊕O. In contrast, Lindel's theorem affirming the BassQuillen conjecture in the geometric case shows that H N i s 1 ( , G L r ) H N i s 1 − , G L r H_(Nis)^(1)(-,GL_(r))\mathrm{H}_{\mathrm{Nis}}^{1}\left(-, \mathrm{GL}_{r}\right)HNis1(−,GLr) is A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant on affines. The next result generalizes this last observation.
Theorem 3.11 (Morel, Schlichting, Asok-Hoyois-Wendt). If X X XXX is a smooth affine k k kkk-scheme, then for any r N r ∈ N r inNr \in \mathbb{N}r∈N there are functorial bijections of the form
π 0 ( Sing A 1 Gr r ( X ) ) [ X , Gr r ] A 1 Vect r ( X ) . Ï€ 0 Sing A 1 ⁡ Gr r ⁡ ( X ) → ∼ X , Gr r A 1 → ∼ Vect r ⁡ ( X ) . pi_(0)(Sing^(A^(1))Gr_(r)(X))rarr"∼"[X,Gr_(r)]_(A^(1))rarr"∼"Vect_(r)(X).\pi_{0}\left(\operatorname{Sing}^{\mathbb{A}^{1}} \operatorname{Gr}_{r}(X)\right) \xrightarrow{\sim}\left[X, \operatorname{Gr}_{r}\right]_{\mathbb{A}^{1}} \xrightarrow{\sim} \operatorname{Vect}_{r}(X) .Ï€0(SingA1⁡Grr⁡(X))→∼[X,Grr]A1→∼Vectr⁡(X).
Remark 3.12. The above result was first established by F. Morel in [40] for r 2 r ≠ 2 r!=2r \neq 2r≠2 and k k kkk an infinite, perfect field, and his proof was partly simplified by M. Schlichting whose argument also established the case r = 2 r = 2 r=2r=2r=2 [51]. The version above is stated in [13].
Remark 3.13. While the functor of isomorphism classes of vector bundles is A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant on smooth affine k k kkk-schemes, even the latter can fail for G-torsors under more general group schemes, e.g., the special orthogonal group scheme S O n S O n SO_(n)\mathrm{SO}_{n}SOn (see [47] or [45]). Furthermore, while G L n G L n GL_(n)\mathrm{GL}_{n}GLn-torsors are always locally trivial with respect to the Nisnevich (and even the Zariski) topology, for an arbitrary smooth k k kkk-group scheme G, one only knows that G-torsors are locally trivial with respect to the étale topology.
In [14,15], it is shown that if G G G\mathrm{G}G is an isotropic reductive group scheme (see [14, DEFINITION 3.3.5] for the definition), then the functor assigning to X Sm k aff X ∈ Sm k aff  X inSm_(k)^("aff ")X \in \operatorname{Sm}_{k}^{\text {aff }}X∈Smkaff  the set H N i s 1 ( X , G ) H N i s 1 ( X , G ) H_(Nis)^(1)(X,G)\mathrm{H}_{\mathrm{Nis}}^{1}(X, \mathrm{G})HNis1(X,G) is representable by BG. This observation has a number of consequences, e.g., the following result about quadrics (see Examples 3.7 and 3.8).
Theorem 3.14 ([1,14,15]). For any integer i 1 i ≥ 1 i >= 1i \geq 1i≥1 and any X S m k aff X ∈ S m k aff  X inSm_(k)^("aff ")X \in \mathrm{Sm}_{k}^{\text {aff }}X∈Smkaff , the comparison map
π 0 ( Sing A 1 Q i ( X ) ) [ X , Q i ] A 1 Ï€ 0 Sing A 1 ⁡ Q i ( X ) → ∼ X , Q i A 1 pi_(0)(Sing^(A^(1))Q_(i)(X))rarr"∼"[X,Q_(i)]_(A^(1))\pi_{0}\left(\operatorname{Sing}^{\mathbb{A}^{1}} Q_{i}(X)\right) \xrightarrow{\sim}\left[X, Q_{i}\right]_{\mathbb{A}^{1}}Ï€0(SingA1⁡Qi(X))→∼[X,Qi]A1
is a bijection, contravariantly functorial in X X XXX.

3.5. Postnikov towers, connectedness and strictly A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant sheaves

Recall from Definition 3.9 the notion of strongly or strictly A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant sheaves of groups. F. Morel showed that such sheaves can be thought of as "building blocks" for the unstable A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy category. Morel's foundational works [ 39 , 40 ] [ 39 , 40 ] [39,40][39,40][39,40] can be viewed as a careful analysis of strictly and strongly A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant sheaves of groups and the relationship between the two notions. More precisely, Morel showed that working over a perfect
field k k kkk, the A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy sheaves of a motivic space are always strongly A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant, and that strongly A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant sheaves of abelian groups are automatically strictly A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant.
To check this, Morel showed that strongly (resp. strictly) A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant sheaves of groups come equipped with a package of results/tools that are central to computations; this package of results is essentially an extension/amalgam/axiomatization of work of BlochOgus and Gabber on étale cohomology exposed in [26] and Rost [50].
Example 3.15. Some examples of A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant sheaves that will appear in the sequel are:
  • unramified Milnor K-theory sheaves K i M , i 0 K i M , i ≥ 0 K_(i)^(M),i >= 0\mathbf{K}_{i}^{M}, i \geq 0KiM,i≥0 (see [50, coRoLLARY 6.5, PRoPOSItION 8.6] where, more generally, it is shown that any Rost cycle module gives rise to a strictly A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant sheaf);
  • the Witt sheaf W W W\mathbf{W}W or unramified powers of the fundamental ideal in the Witt ring I j I j I^(j)\mathbf{I}^{j}Ij, j 0 j ≥ 0 j >= 0j \geq 0j≥0 (this follows from [46]); and
  • unramified Milnor-Witt K-theory sheaves K i M W , i Z K i M W , i ∈ Z K_(i)^(MW),i inZ\mathbf{K}_{i}^{M W}, i \in \mathbb{Z}KiMW,i∈Z (see [40, cHAPTER 3] for this assertion, or [31, COROLLARY 8.5,PROPOSITION 9.1] where this observation is generalized to so-called Milnor-Witt cycle modules).
3.16 (Moore-Postnikov factorizations). There is an analog of the Moore-Postnikov factorization of a map f : E B f : E → B f:ErarrBf: \mathscr{E} \rightarrow \mathscr{B}f:E→B of spaces along the lines described in Section 2. For concreteness we discuss the case where E E E\mathscr{E}E and B B B\mathscr{B}B are A 1 A 1 A^(1)\mathbb{A}^{1}A1-connected and f f fff induces an isomorphism on A 1 A 1 A^(1)\mathbb{A}^{1}A1-fundamental sheaves of groups for some choice of base-point in E E E\mathscr{E}E.
Given f f fff as above, there are τ i f Spc k Ï„ ≤ i f ∈ Spc k tau_( <= i)f inSpc_(k)\tau_{\leq i} f \in \operatorname{Spc}_{k}τ≤if∈Spck together with maps E τ i f , τ i f B E → Ï„ ≤ i f , Ï„ ≤ i f → B Erarrtau_( <= i)f,tau_( <= i)f rarrB\mathscr{E} \rightarrow \tau_{\leq i} f, \tau_{\leq i} f \rightarrow \mathscr{B}E→τ≤if,τ≤if→B and τ i f τ i 1 f Ï„ ≤ i f → Ï„ ≤ i − 1 f tau_( <= i)f rarrtau_( <= i-1)f\tau_{\leq i} f \rightarrow \tau_{\leq i-1} fτ≤if→τ≤i−1f fitting into a diagram of exactly the same form as (2.1) (replacing E E EEE by E E E\mathscr{E}E and B B BBB by B ) B ) B)\mathscr{B})B). The relevant properties of this presentation are similar to those sketched before (replacing homotopy groups by homotopy sheaves), together with a homotopy pullback diagram of exactly the same form as (2.2). We refer to this tower as the A 1 A 1 A^(1)\mathbb{A}^{1}A1-Moore-Postnikov tower of f f fff and the reader may consult [40, APPENDIX B] or [5, §6] for a more detailed presentation.
If X X XXX is a smooth scheme, then a map ψ : X B ψ : X → B psi:X rarrB\psi: X \rightarrow \mathscr{B}ψ:X→B lifts to ψ ~ : X E ψ ~ : X → E tilde(psi):X rarrE\tilde{\psi}: X \rightarrow \mathscr{E}ψ~:X→E if and only if lifts exist at each stage of the tower, i.e., if and only if a suitable obstruction vanishes. These obstructions are, by construction, valued in Nisnevich cohomology on X X XXX with values in a strictly A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant sheaf (see [ 5 , § 6 ] [ 5 , § 6 ] [5,§6][5, \S 6][5,§6] for a more detailed explanation).
By analogy with the situation in topology, we will use the A 1 A 1 A^(1)\mathbb{A}^{1}A1-Moore-Postnikov factorization to study lifting problems by means of obstruction theory. The relevant obstructions will lie in cohomology groups of a smooth scheme with coefficients in a strictly A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant sheaf. This motivates the following definition.
Definition 3.17. Let X X XXX be a smooth k k kkk-scheme. We say that X X XXX has A 1 A 1 A^(1)\mathbb{A}^{1}A1-cohomological dimension d ≤ d <= d\leq d≤d if for any integer i > d i > d i > di>di>d and any strictly A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant sheaf F , H N i s i ( X , F ) = 0 F , H N i s i ( X , F ) = 0 F,H_(Nis)^(i)(X,F)=0\mathbf{F}, \mathrm{H}_{\mathrm{Nis}}^{i}(X, \mathbf{F})=0F,HNisi(X,F)=0. In that case, we write c d A 1 ( X ) d c d A 1 ( X ) ≤ d cd_(A^(1))(X) <= dc d_{\mathbb{A}^{1}}(X) \leq dcdA1(X)≤d.
Example 3.18. If X X XXX is a smooth k k kkk-scheme of dimension d d ddd, then X X XXX necessarily has A 1 A 1 A^(1)\mathbb{A}^{1}A1-cohomological dimension d ≤ d <= d\leq d≤d as well. Since A n A n A^(n)\mathbb{A}^{n}An has A 1 A 1 A^(1)\mathbb{A}^{1}A1-cohomological dimension 0 ≤ 0 <= 0\leq 0≤0, the A 1 A 1 A^(1)\mathbb{A}^{1}A1-cohomological dimension can be strictly smaller than Krull dimension; Example 3.6 gives numerous other such examples.

3.6. Complex realization

Assume k k kkk is a field that admits an embedding ι C : k C ι C : k ↪ C iotaC:k↪C\iota \mathbb{C}: k \hookrightarrow \mathbb{C}ιC:k↪C. The functor that assigns to a smooth k k kkk-variety X X XXX the complex manifold X ( C ) X ( C ) X(C)X(\mathbb{C})X(C) equipped with its classical topology extends to a complex realization functor
C : H ( k ) H ℜ C : H ( k ) → H ℜ_(C):H(k)rarrH\Re_{\mathbb{C}}: \mathrm{H}(k) \rightarrow \mathrm{H}ℜC:H(k)→H
where H H H\mathrm{H}H is the usual homotopy category of topological spaces [41, §3.3]. By construction, complex realization preserves finite products and homotopy colimits. It follows that the complex realization of the motivic sphere S p , q S p , q S^(p,q)S^{p, q}Sp,q is the ordinary sphere S p + q S p + q S^(p+q)S^{p+q}Sp+q, and consequently the complex realization functor induces group homomorphisms of the form
π i , j A ( X , x ) ( C ) π i + j ( X ( C ) , x ) Ï€ i , j A ( X , x ) ( C ) → Ï€ i + j ( X ( C ) , x ) pi_(i,j)^(A)(X,x)(C)rarrpi_(i+j)(X(C),x)\pi_{i, j}^{\mathbb{A}}(X, x)(\mathbb{C}) \rightarrow \pi_{i+j}(X(\mathbb{C}), x)Ï€i,jA(X,x)(C)→πi+j(X(C),x)
for any pointed smooth k k kkk-scheme ( X , x ) ( X , x ) (X,x)(X, x)(X,x).
Suppose X X XXX is any k k kkk-scheme admitting a complex embedding and fix such an embedding. Write Vect r top ( X ) Vect r top  ⁡ ( X ) Vect_(r)^("top ")(X)\operatorname{Vect}_{r}^{\text {top }}(X)Vectrtop ⁡(X) for the set of isomorphism classes of complex topological vector bundles on X X XXX. There is a function
Vect r ( X ) Vect r t o p ( X ) Vect r ⁡ ( X ) → Vect r t o p ⁡ ( X ) Vect_(r)(X)rarrVect_(r)^(top)(X)\operatorname{Vect}_{r}(X) \rightarrow \operatorname{Vect}_{r}^{\mathrm{top}}(X)Vectr⁡(X)→Vectrtop⁡(X)
sending an algebraic vector bundle E E EEE over X X XXX to the topological vector bundle on X ( C ) X ( C ) X(C)X(\mathbb{C})X(C) attached to the base change of E E EEE to X C X C X_(C)X_{\mathbb{C}}XC. We will say that an algebraic vector bundle is algebraizable if it lies in the image of this map.
As rank r r rrr topological vector bundles are classified by the set [ X ( C ) , B U ( r ) ] [ X ( C ) , B U ( r ) ] [X(C),BU(r)][X(\mathbb{C}), \mathrm{B} U(r)][X(C),BU(r)] of homotopy classes of maps from X ( C ) X ( C ) X(C)X(\mathbb{C})X(C) to the complex Grassmannian, it follows that the function of the preceding paragraph factors as
Vect r ( X ) [ X , Gr r ] A 1 Vect r top ( X ) Vect r ⁡ ( X ) → X , Gr r A 1 → Vect r top ⁡ ( X ) Vect_(r)(X)rarr[X,Gr_(r)]_(A^(1))rarrVect_(r)^(top)(X)\operatorname{Vect}_{r}(X) \rightarrow\left[X, \operatorname{Gr}_{r}\right]_{\mathbb{A}^{1}} \rightarrow \operatorname{Vect}_{r}^{\operatorname{top}}(X)Vectr⁡(X)→[X,Grr]A1→Vectrtop⁡(X)
Theorem 3.11 implies that the first map is a bijection if X X XXX is a smooth affine k k kkk-scheme (or, alternatively, if r = 1 r = 1 r=1r=1r=1 ). More generally, combining Theorem 3.11 and Lemma 3.3 one knows that any element of [ X , G r r ] A 1 X , G r r A 1 [X,Gr_(r)]_(A^(1))\left[X, \mathrm{Gr}_{r}\right]_{\mathbb{A}^{1}}[X,Grr]A1 may be represented by an actual rank r r rrr vector bundle on any Jouanolou device X ~ X ~ tilde(X)\tilde{X}X~ of X X XXX; this suggests the following definition.
Definition 3.19. If X X XXX is a smooth k k kkk-scheme, then by a rank r r rrr motivic vector bundle on X X XXX we mean an element of the set [ X , G r r ] A 1 X , G r r A 1 [X,Gr_(r)]_(A^(1))\left[X, \mathrm{Gr}_{r}\right]_{\mathbb{A}^{1}}[X,Grr]A1.
Question 3.20. If X X XXX is a smooth complex algebraic variety, then which topological vector bundles are algebraizable (resp. motivic)?

4. OBSTRUCTION THEORY AND VECTOR BUNDLES

In order to apply the obstruction theory described in the previous sections to analyze algebraic vector bundles, we need more information about the structure of the classifying space B G L n B G L n BGL_(n)\mathrm{BGL}_{n}BGLn including information about its A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy sheaves, and the structure of the homotopy fiber of the stabilization map B G L n B G L n + 1 B G L n → B G L n + 1 BGL_(n)rarrBGL_(n+1)\mathrm{BGL}_{n} \rightarrow \mathrm{BGL}_{n+1}BGLn→BGLn+1 induced by the map G L n G L n + 1 G L n → G L n + 1 GL_(n)rarrGL_(n+1)\mathrm{GL}_{n} \rightarrow \mathrm{GL}_{n+1}GLn→GLn+1 sending an invertible matrix X X XXX to the block matrix diag ( 1 , X ) diag ⁡ ( 1 , X ) diag(1,X)\operatorname{diag}(1, X)diag⁡(1,X).

4.1. The homotopy sheaves of the classifying space of B G L n B G L n BGL_(n)\mathrm{BGL}_{\boldsymbol{n}}BGLn

We observed earlier that B G L 1 = B G m B G L 1 = B G m BGL_(1)=BG_(m)\mathrm{BGL}_{1}=\mathrm{B} \mathbb{G}_{m}BGL1=BGm is an Eilenberg-Mac Lane space for the sheaf G m G m G_(m)\mathbb{G}_{m}Gm : it is A 1 A 1 A^(1)\mathbb{A}^{1}A1-connected, and has exactly 1 nonvanishing A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy sheaf in degree 1 , which is isomorphic to G m G m G_(m)\mathbb{G}_{m}Gm. For n 1 n ≥ 1 n >= 1n \geq 1n≥1, the analysis of homotopy sheaves of B G L n B G L n BGL_(n)\mathrm{BGL}_{n}BGLn uses several ingredients. First, Morel-Voevodsky observed that B G L = colim n B G L n B G L = colim n ⁡ B G L n BGL=colim_(n)BGL_(n)\mathrm{BGL}=\operatorname{colim}_{n} \mathrm{BGL}_{n}BGL=colimn⁡BGLn (for the inclusions described above) represents (reduced) algebraic K-theory after [41, 84 THEOREM 3.13]. Second, Morel observed that there is an A 1 A 1 A^(1)\mathbb{A}^{1}A1-fiber sequence of the form
(4.1) A n + 1 0 B G L n B G L n + 1 (4.1) A n + 1 ∖ 0 → B G L n → B G L n + 1 {:(4.1)A^(n+1)\\0rarrBGL_(n)rarrBGL_(n+1):}\begin{equation*} \mathbb{A}^{n+1} \backslash 0 \rightarrow \mathrm{BGL}_{n} \rightarrow \mathrm{BGL}_{n+1} \tag{4.1} \end{equation*}(4.1)An+1∖0→BGLn→BGLn+1
and that A n + 1 0 A n + 1 ∖ 0 A^(n+1)\\0\mathbb{A}^{n+1} \backslash 0An+1∖0 is A 1 ( n 1 ) A 1 − ( n − 1 ) A^(1)-(n-1)\mathbb{A}^{1}-(n-1)A1−(n−1)-connected. Furthermore, Morel computed [40] the first nonvanishing A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy sheaf of A n + 1 0 A n + 1 ∖ 0 A^(n+1)\\0\mathbb{A}^{n+1} \backslash 0An+1∖0 in terms of what he called Milnor-Witt K-theory sheaves (Example 3.15).
Putting these ingredients together, one deduces
π i A 1 ( B G L n ) K i Q , 1 i n 1 Ï€ i A 1 B G L n ≅ K i Q , 1 ≤ i ≤ n − 1 pi_(i)^(A^(1))(BGL_(n))~=K_(i)^(Q),quad1 <= i <= n-1\pi_{i}^{\mathbb{A}^{1}}\left(\mathrm{BGL}_{n}\right) \cong \mathbf{K}_{i}^{Q}, \quad 1 \leq i \leq n-1Ï€iA1(BGLn)≅KiQ,1≤i≤n−1
where K i Q K i Q K_(i)^(Q)\mathbf{K}_{i}^{Q}KiQ is the (Nisnevich) sheafification of the Quillen K-theory presheaf on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk. Following terminology from topology, sheaves in this range are called stable, and the case i = n i = n i=ni=ni=n is called the first unstable homotopy sheaf. In [3], we described the first unstable homotopy sheaf of B G L n B G L n BGL_(n)\mathrm{BGL}_{n}BGLn.
The group scheme G L n G L n GL_(n)\mathrm{GL}_{n}GLn maps to G L n ( C ) G L n ( C ) GL_(n)(C)\mathrm{GL}_{n}(\mathbb{C})GLn(C) under complex realization; the latter is homotopy equivalent to U ( n ) U ( n ) U(n)U(n)U(n). For context, we recall some facts about homotopy of U ( n ) U ( n ) U(n)U(n)U(n). A classical result of Bott, refining results of Borel-Hirzeburch [24, THEOREM 25.8] asserts that the image of π 2 n ( B U ( n ) ) Ï€ 2 n ( B U ( n ) ) pi_(2n)(BU(n))\pi_{2 n}(B U(n))Ï€2n(BU(n)) in H 2 n ( B U ( n ) ) H 2 n ( B U ( n ) ) H_(2n)(BU(n))H_{2 n}(B U(n))H2n(BU(n)) is divisible by precisely ( n 1 ) ( n − 1 ) (n-1)(n-1)(n−1) ! [25]. This result implies the assertion that π 2 n ( U ( n ) ) = n Ï€ 2 n ( U ( n ) ) = n pi_(2n)(U(n))=n\pi_{2 n}(U(n))=nÏ€2n(U(n))=n !.
Complex realization yields a map π n , n A 1 ( G L n ) π 2 n ( U ( n ) ) Ï€ n , n A 1 G L n → Ï€ 2 n ( U ( n ) ) pi_(n,n)^(A^(1))(GL_(n))rarrpi_(2n)(U(n))\pi_{n, n}^{\mathbb{A}^{1}}\left(\mathrm{GL}_{n}\right) \rightarrow \pi_{2 n}(U(n))Ï€n,nA1(GLn)→π2n(U(n)). One can view the celebrated "Suslin matrices" [59] as providing an algebro-geometric realization of the generator of π 2 n ( U ( n ) ) Ï€ 2 n ( U ( n ) ) pi_(2n)(U(n))\pi_{2 n}(U(n))Ï€2n(U(n)). Analyzing the fiber sequence of (4.1) and putting all of the ingredients above together, we obtain the following result (we refer the reader to Example 3.15 for notation).
Theorem 4.1 ([3, THEOREM 1.1]). Assume k k kkk is a field that has characteristic not equal to 2. For any integer n 2 n ≥ 2 n >= 2n \geq 2n≥2, there are strictly A 1 A 1 A^(1)\mathbb{A}^{1}A1-invariant sheaves S n S n S_(n)\mathbf{S}_{n}Sn fitting into exact sequences of the form:
0 S n + 1 π n A 1 ( B G L n ) K n Q 0 , oodd ; 0 S n + 1 × K n + 1 M / 2 I n + 1 π n A 1 ( B G L n ) K n Q 0 , n even 0 → S n + 1 → Ï€ n A 1 B G L n → K n Q → 0 ,  oodd  ; 0 → S n + 1 × K n + 1 M / 2 I n + 1 → Ï€ n A 1 B G L n → K n Q → 0 ,  n even  {:[0rarrS_(n+1),rarrpi_(n)^(A^(1))(BGL_(n)),rarrK_(n)^(Q)rarr0",",],[," oodd ";],[0rarrS_(n+1)xx_(K_(n+1)^(M)//2)I^(n+1),rarrpi_(n)^(A^(1))(BGL_(n)),rarrK_(n)^(Q)rarr0","," n even "]:}\begin{array}{rlrl} 0 \rightarrow \mathbf{S}_{n+1} & \rightarrow \pi_{n}^{\mathbb{A}^{1}}\left(\mathrm{BGL}_{n}\right) & \rightarrow \mathbf{K}_{n}^{Q} \rightarrow 0, & \\ & \text { oodd } ; \\ 0 \rightarrow \mathbf{S}_{n+1} \times_{\mathbf{K}_{n+1}^{M} / 2} \mathbf{I}^{n+1} & \rightarrow \pi_{n}^{\mathbb{A}^{1}}\left(\mathrm{BGL}_{n}\right) & \rightarrow \mathbf{K}_{n}^{Q} \rightarrow 0, & \text { n even } \end{array}0→Sn+1→πnA1(BGLn)→KnQ→0, oodd ;0→Sn+1×Kn+1M/2In+1→πnA1(BGLn)→KnQ→0, n even 
where
(1) there is a canonical epimorphism K n M / ( n 1 ) ! S n K n M / ( n − 1 ) ! → S n K_(n)^(M)//(n-1)!rarrS_(n)\mathbf{K}_{n}^{M} /(n-1)!\rightarrow \mathbf{S}_{n}KnM/(n−1)!→Sn which becomes an isomorphism after n 2 n − 2 n-2n-2n−2 contractions (see [ 3 , § 2 . 3 ] [ 3 , § 2 . 3 ] [3,§2.3][3, \S \mathbf{2} . \mathbf{3}][3,§2.3] for this terminology);
(2) there is a canonical epimorphism S n K n M / 2 S n → K n M / 2 S_(n)rarrK_(n)^(M)//2\mathbf{S}_{n} \rightarrow \mathbf{K}_{n}^{M} / 2Sn→KnM/2 such that the composite
K n M / ( n 1 ) ! S n K n M / 2 K n M / ( n − 1 ) ! → S n → K n M / 2 K_(n)^(M)//(n-1)!rarrS_(n)rarrK_(n)^(M)//2\mathbf{K}_{n}^{M} /(n-1)!\rightarrow \mathbf{S}_{n} \rightarrow \mathbf{K}_{n}^{M} / 2KnM/(n−1)!→Sn→KnM/2
is reduction modulo 2 ;
(3) the fiber product is taken over the epimorphism S n + 1 K n + 1 M / 2 S n + 1 → K n + 1 M / 2 S_(n+1)rarrK_(n+1)^(M)//2\mathbf{S}_{n+1} \rightarrow \mathbf{K}_{n+1}^{M} / 2Sn+1→Kn+1M/2 and a sheafified version of Milnor's homomorphism I n + 1 K n + 1 M / 2 I n + 1 → K n + 1 M / 2 I^(n+1)rarrK_(n+1)^(M)//2\mathbf{I}^{n+1} \rightarrow \mathbf{K}_{n+1}^{M} / 2In+1→Kn+1M/2.
Moreover, if k k kkk admits a complex embedding, then the map
π n , n + 1 A 1 ( B G L n ) ( C ) π 2 n + 1 ( B U ( n ) ) Z / n ! Ï€ n , n + 1 A 1 B G L n ( C ) → Ï€ 2 n + 1 ( B U ( n ) ) ≅ Z / n ! pi_(n,n+1)^(A^(1))(BGL_(n))(C)rarrpi_(2n+1)(BU(n))~=Z//n!\pi_{n, n+1}^{\mathbb{A}^{1}}\left(\mathrm{BGL}_{n}\right)(\mathbb{C}) \rightarrow \pi_{2 n+1}(B U(n)) \cong \mathbb{Z} / n!Ï€n,n+1A1(BGLn)(C)→π2n+1(BU(n))≅Z/n!
induced by complex realization is an isomorphism.
Bott's refinement of the theorem of Borel-Hirzebruch turns out to have an algebrogeometric interpretation. Indeed, in joint work with T. B. Williams [12] we showed that S n S n S_(n)\mathbf{S}_{n}Sn can described using a "Hurewicz map" analyzed by Andrei Suslin [60]. Suslin's conjecture on the image of this map is equivalent to the following conjecture.
Conjecture 4.2 (Suslin's factorial conjecture). The canonical epimorphism K n M K n M K_(n)^(M)\mathbf{K}_{n}^{M}KnM / ( n 1 ) ! S n ( n − 1 ) ! → S n (n-1)!rarrS_(n)(n-1)!\rightarrow \mathbf{S}_{n}(n−1)!→Sn is an isomorphism.
Remark 4.3. The conjecture holds tautologically for n = 2 n = 2 n=2n=2n=2. For n = 3 n = 3 n=3n=3n=3, Suslin observed the conjecture was equivalent to the Milnor conjecture on quadratic forms, which was resolved later independently by Merkurjev-Suslin and Rost. The conjecture was established for n = 5 n = 5 n=5n=5n=5 in "most" cases in [12] (see the latter for a precise statement); this work relies heavily on the computation by Østvær-Röndigs-Spitzweck of the motivic stable 1-stem [49].

4.2. Splitting bundles, Euler classes, and cohomotopy

Morel's computations around A n 0 A n ∖ 0 A^(n)\\0\mathbb{A}^{n} \backslash 0An∖0 in conjunction with the fiber sequences of (4.1) allow a significant improvement of Serre's celebrated splitting theorem for smooth affine varieties over a field that we stated in the introduction.
Proof of the motivic Serre Splitting Theorem 1.1. Suppose X X XXX is a smooth affine k k kkk-variety having A 1 A 1 A^(1)\mathbb{A}^{1}A1-cohomological dimension d ≤ d <= d\leq d≤d, and suppose ξ : X B G L r ξ : X → B G L r xi:X rarrBGL_(r)\xi: X \rightarrow \mathrm{BGL}_{r}ξ:X→BGLr classifies a rank r > d r > d r > dr>dr>d vector bundle on X X XXX. We proceed by analyzing the A 1 A 1 A^(1)\mathbb{A}^{1}A1-Moore-Postnikov factorization of the stabilization map (4.1) with n = r 1 n = r − 1 n=r-1n=r-1n=r−1. In that case, combining the fact that A r 0 A r ∖ 0 A^(r)\\0\mathbb{A}^{r} \backslash 0Ar∖0 is A 1 A 1 A^(1)\mathbb{A}^{1}A1 - ( r 2 ) ( r − 2 ) (r-2)(r-2)(r−2)-connected and the A 1 A 1 A^(1)\mathbb{A}^{1}A1-cohomological dimension assumption on X X XXX, one sees all obstructions to splitting vanish.
Remark 4.4. The proof of this result does not rely on the Serre splitting theorem. Since A 1 A 1 A^(1)\mathbb{A}^{1}A1-cohomological dimension can be strictly smaller than Krull dimension (Example 3.18), this statement is strictly stronger than Serre splitting. Importantly, the improvement achieved here seems inaccessible to classical techniques.
The algebro-geometric splitting problem in corank 0 on smooth affine varieties of dimension d d ddd over a field k k kkk has been analyzed by many authors. When k k kkk is an algebraically closed field, M. P. Murthy proved that the top Chern class in Chow groups is the only obstruction to splitting [42]. When k k kkk is not algebraically closed, vanishing of the top Chern class is known to be insufficient to guarantee splitting, and Nori proposed some ideas to analyze this situation. His ideas led Bhatwadekar and Sridharan [23] to introduce what they called Euler class groups and to provide one explicit "generators and relations" answer to this question. At the same time, F. Morel proposed an approach to the splitting problem in corank 0 , which we recall here.
Theorem 4.5 (Morel's splitting theorem [40, THEOREM 1.32]). Assume k k kkk is a field and X X XXX is a smooth affine k k kkk-variety of A 1 A 1 A^(1)\mathbb{A}^{1}A1-cohomological dimension d ≤ d <= d\leq d≤d. If E E E\mathcal{E}E is a rank d d ddd vector bundle on X X XXX, then E E E\mathcal{E}E splits off a free rank 1 summand if and only if an Euler class e ( E ) H N i s d ( X , K d M W ( det E ) ) e ( E ) ∈ H N i s d X , K d M W ( det ⁡ E ) e(E)inH_(Nis)^(d)(X,K_(d)^(MW)(det E))e(\mathcal{E}) \in \mathrm{H}_{\mathrm{Nis}}^{d}\left(X, \mathbf{K}_{d}^{M W}(\operatorname{det} \mathcal{E})\right)e(E)∈HNisd(X,KdMW(det⁡E)) vanishes.
Remark 4.6. The Euler class of Theorem 4.5 is precisely the first nonvanishing obstruction class, as described in Paragraph 3.16. A related "cohomological" approach to the splitting problem in corank 0 was proposed by Barge-Morel [20] and analyzed in the thesis of the second author [29]. The cohomological approach was in most cases shown to be equivalent to the "obstruction-theoretic" approach in [6]. We also refer the reader to [51] for related results on the theory of Euler classes, extending also to singular varieties.
The next result shows that the relationship between Euler classes à la BhatwadekarSridharan and Euler classes à la Morel is mediated by another topologically inspired notion: cohomotopy (at least for bundles of trivial determinant).
Theorem 4.7 ([8, тheorem 1]). Suppose k k kkk is a field, n n nnn and d d ddd are integers, n 2 n ≥ 2 n >= 2n \geq 2n≥2, and X X XXX is a smooth affine k k kkk-scheme of dimension d 2 n 2 d ≤ 2 n − 2 d <= 2n-2d \leq 2 n-2d≤2n−2. Write E n ( X ) E n ( X ) E^(n)(X)\mathrm{E}^{n}(X)En(X) for the BhatwadekarSridharan Euler class group.
  • The set [ X , Q 2 n ] A 1 X , Q 2 n A 1 [X,Q_(2n)]_(A^(1))\left[X, Q_{2 n}\right]_{\mathbb{A}^{1}}[X,Q2n]A1 carries a functorial abelian group structure;
  • There are functorial homomorphisms:
E n ( X ) s [ X , Q 2 n ] A 1 h H N i s d ( X , K n M W ) E n ( X ) → s X , Q 2 n A 1 → h H N i s d X , K n M W E^(n)(X)rarr"s"[X,Q_(2n)]_(A^(1))rarr"h"H_(Nis)^(d)(X,K_(n)^(MW))\mathrm{E}^{n}(X) \xrightarrow{s}\left[X, Q_{2 n}\right]_{\mathbb{A}^{1}} \xrightarrow{h} \mathrm{H}_{\mathrm{Nis}}^{d}\left(X, \mathbf{K}_{n}^{M W}\right)En(X)→s[X,Q2n]A1→hHNisd(X,KnMW)
where the "Segre class" homomorphism s is surjective and an isomorphism if k k kkk is infinite and d 2 d ≥ 2 d >= 2d \geq 2d≥2, and the Hurewicz homomorphism h h hhh is an isomorphism if d n d ≤ n d <= nd \leq nd≤n.
Remark 4.8. The group structure on [ X , Q 2 n ] A 1 X , Q 2 n A 1 [X,Q_(2n)]_(A^(1))\left[X, Q_{2 n}\right]_{\mathbb{A}^{1}}[X,Q2n]A1 is an algebro-geometric variant of Borsuk's group structure on cohomotopy. The second point of the statement includes the algebrogeometric analog of the Hopf classification theorem from topology.

4.3. The next nontrivial A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy sheaf of spheres

In Section 2.2 we described a cohomological approach to the splitting problem in corank 1 for smooth closed manifolds of dimension d d ddd; this approach relied on the computa-
tion of π d ( S d 1 ) Ï€ d S d − 1 pi_(d)(S^(d-1))\pi_{d}\left(S^{d-1}\right)Ï€d(Sd−1). In order to analyze the algebro-geometric splitting problem in corank 1 using the A 1 A 1 A^(1)\mathbb{A}^{1}A1-Moore-Postnikov factorization we will need as input further information about the homotopy sheaves of A d 0 A d ∖ 0 A^(d)\\0\mathbb{A}^{d} \backslash 0Ad∖0. We now describe known results in this direction. For technical reasons, we assume 2 is invertible in what follows.

4.3.1. The K O K O KOK OKO-degree map

In classical algebraic topology, all of the "low degree" elements in the homotopy of spheres can be realized by constructions of "linear algebraic" nature. The situation in algebraic geometry appears to be broadly similar. The first contribution to the "next" nontrivial homotopy sheaves of motivic spheres requires recalling the geometric formulation of Bott periodicity for Hermitian K-theory given by Schlichting-Tripathi.
We write O O O\mathrm{O}O for the infinite orthogonal group. In topology, Bott periodicity identifies the 8 -fold loop space of O O O\mathrm{O}O with itself and identifies the intermediate loop spaces in concrete geometric terms. In algebraic geometry, Schlichting and Tripathi proved that the 4-fold P 1 P 1 P^(1)\mathbb{P}^{1}P1-loop space Ω P 1 4 O Ω P 1 4 O Omega_(P^(1))^(4)O\Omega_{\mathbb{P}^{1}}^{4} \mathrm{O}ΩP14O also coincides with O O O\mathrm{O}O and realized suitable intermediate loop spaces: Ω P 1 n O Ω P 1 n O Omega_(P^(1))^(n)O\Omega_{\mathbb{P}^{1}}^{n} \mathrm{O}ΩP1nO is isomorphic to GL / O when n = 1 , S p n = 1 , S p n=1,Spn=1, \mathrm{Sp}n=1,Sp when n = 2 n = 2 n=2n=2n=2 and GL / S p / S p //Sp/ \mathrm{Sp}/Sp when n = 3 n = 3 n=3n=3n=3, where S p S p Sp\mathrm{Sp}Sp is the stable sympletic group, GL/O is the ind-variety of invertible symmetric matrices, and GL/Sp is the ind-variety of invertible skew-symmetric matrices [52,

THEOREMS 8.2 AND 8.4].

A slight modification of the Suslin matrix construction [59, LEMMA 5.3] yields a map
u n : Q 2 n 1 Ω P 1 n O u n : Q 2 n − 1 → Ω P 1 − n O u_(n):Q_(2n-1)rarrOmega_(P^(1))^(-n)Ou_{n}: Q_{2 n-1} \rightarrow \Omega_{\mathbb{P}^{1}}^{-n} Oun:Q2n−1→ΩP1−nO
called the (unstable) KO-degree map in weight n n nnn that was analyzed in detail in [7]. The terminology stems from the fact that this map stabilizes to the "unit map from the sphere spectrum to the Hermitian K-theory spectrum" in an appropriate sense. The scheme Q 2 n 1 Q 2 n − 1 Q_(2n-1)Q_{2 n-1}Q2n−1 is A 1 A 1 A^(1)\mathbb{A}^{1}A1 - ( n 2 ) ( n − 2 ) (n-2)(n-2)(n−2)-connected by combining the weak equivalence of Example 3.7 and Morel's connectivity results for A n 0 A n ∖ 0 A^(n)\\0\mathbb{A}^{n} \backslash 0An∖0. Thus, u n u n u_(n)u_{n}un factors through the A 1 ( n 2 ) A 1 − ( n − 2 ) A^(1)-(n-2)\mathbb{A}^{1}-(n-2)A1−(n−2)-connected cover of Ω P 1 n O Ω P 1 − n O Omega_(P^(1))^(-n)O\Omega_{\mathbb{P}^{1}}^{-n} OΩP1−nO.
Taking homotopy sheaves on both sides, there are induced morphisms
π i A 1 ( u ) : π i A 1 ( Q 2 n 1 ) π i A 1 ( Ω P 1 n O ) Ï€ i A 1 ( u ) : Ï€ i A 1 Q 2 n − 1 → Ï€ i A 1 Ω P 1 − n O pi_(i)^(A^(1))(u):pi_(i)^(A^(1))(Q_(2n-1))rarrpi_(i)^(A^(1))(Omega_(P^(1))^(-n)O)\pi_{i}^{\mathbb{A}^{1}}(u): \pi_{i}^{\mathbb{A}^{1}}\left(Q_{2 n-1}\right) \rightarrow \pi_{i}^{\mathbb{A}^{1}}\left(\Omega_{\mathbb{P}^{1}}^{-n} O\right)Ï€iA1(u):Ï€iA1(Q2n−1)→πiA1(ΩP1−nO)
This homomorphism is trivial if i < n 1 i < n − 1 i < n-1i<n-1i<n−1 by connectivity estimates. If i = n 1 i = n − 1 i=n-1i=n-1i=n−1, via Morel's calculations one obtains a morphism K n M W G W n n K n M W → G W n n K_(n)^(MW)rarrGW_(n)^(n)\mathbf{K}_{n}^{\mathrm{MW}} \rightarrow \mathbf{G W}_{n}^{n}KnMW→GWnn whose sections over finitely generated field extensions of k k kkk can be viewed as a quadratic enhancement of the "natural" map from Milnor K K KKK-theory to Quillen K K KKK-theory defined by symbols; we will refer to it as the natural homomorphism (the natural homomorphism is known to be an isomorphism if n 4 n ≤ 4 n <= 4n \leq 4n≤4; the case n 2 n ≤ 2 n <= 2n \leq 2n≤2 is essentially Suslin's, n = 3 n = 3 n=3n=3n=3 is [7, THEOREM 4.3.1], and n = 4 n = 4 n=4n=4n=4 is unpublished work of O. Röndigs).
When i = n i = n i=ni=ni=n, we obtain a morphism
π n A 1 ( A n 0 ) π n A 1 ( Q 2 n 1 ) π n A 1 ( Ω P 1 n O ) G W n + 1 n Ï€ n A 1 A n ∖ 0 ≅ Ï€ n A 1 Q 2 n − 1 → Ï€ n A 1 Ω P 1 − n O ≅ G W n + 1 n pi_(n)^(A^(1))(A^(n)\\0)~=pi_(n)^(A^(1))(Q_(2n-1))rarrpi_(n)^(A^(1))(Omega_(P^(1))^(-n)O)~=GW_(n+1)^(n)\pi_{n}^{\mathbb{A}^{1}}\left(\mathbb{A}^{n} \backslash 0\right) \cong \pi_{n}^{\mathbb{A}^{1}}\left(Q_{2 n-1}\right) \rightarrow \pi_{n}^{\mathbb{A}^{1}}\left(\Omega_{\mathbb{P}^{1}}^{-n} O\right) \cong \mathbf{G} \mathbf{W}_{n+1}^{n}Ï€nA1(An∖0)≅πnA1(Q2n−1)→πnA1(ΩP1−nO)≅GWn+1n
where the right-hand term is by definition a higher Grothendieck-Witt sheaf (obtained by sheafifying the corresponding higher Grothendieck-Witt presheaf on S m k S m k Sm_(k)\mathrm{Sm}_{k}Smk ). The above map
is an epimorphism for n = 2 , 3 n = 2 , 3 n=2,3n=2,3n=2,3 and it follows from these observations that the morphism is an epimorphism after ( n 3 ) ( n − 3 ) (n-3)(n-3)(n−3) contractions [7, THEOREM 4.4.5].

4.3.2. The motivic J-homomorphism

The classical J-homomorphism has an algebro-geometric counterpart that yields the second contribution to the "next" homotopy sheaf of motivic spheres. The standard action of S L n S L n SL_(n)\mathrm{SL}_{n}SLn on A n A n A^(n)\mathbb{A}^{n}An extends to an action on the one-point compactification P n / P n 1 P n / P n − 1 P^(n)//P^(n-1)\mathbb{P}^{n} / \mathbb{P}^{n-1}Pn/Pn−1. The latter space is a motivic sphere P 1 n P 1 ∧ n P^(1^(^^n))\mathbb{P}^{1^{\wedge n}}P1∧n and thus one obtains a map
Σ P 1 n S L n P 1 n Σ P 1 n S L n → P 1 ∧ n Sigma_(P^(1))^(n)SL_(n)rarrP^(1^(^^n))\Sigma_{\mathbb{P}^{1}}^{n} S L_{n} \rightarrow \mathbb{P}^{1^{\wedge n}}ΣP1nSLn→P1∧n
As S L n S L n SL_(n)\mathrm{SL}_{n}SLn is A 1 A 1 A^(1)\mathbb{A}^{1}A1-connected, it follows that Σ P 1 n S L n Σ P 1 n S L n Sigma_(P^(1))^(n)SL_(n)\Sigma_{\mathbb{P}^{1}}^{n} \mathrm{SL}_{n}ΣP1nSLn is A 1 A 1 A^(1)\mathbb{A}^{1}A1 - n n nnn-connected.
The first nonvanishing A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy sheaf appears in degree n + 1 n + 1 n+1n+1n+1; for n = 2 n = 2 n=2n=2n=2, it is isomorphic to K 4 M W K 4 M W K_(4)^(MW)\mathbf{K}_{4}^{M W}K4MW, while for n 3 n ≥ 3 n >= 3n \geq 3n≥3 it is isomorphic to K n + 2 M K n + 2 M K_(n+2)^(M)\mathbf{K}_{n+2}^{M}Kn+2M; this follows from A 1 A 1 A^(1)\mathbb{A}^{1}A1-Hurewicz theorem combined with [17, PROPOSItIoN 3.3.9] using the fact that π 1 A 1 ( S L n ) = K 2 M Ï€ 1 A 1 S L n = K 2 M pi_(1)^(A^(1))(SL_(n))=K_(2)^(M)\pi_{1}^{\mathbb{A}^{1}}\left(\mathrm{SL}_{n}\right)=\mathbf{K}_{2}^{M}Ï€1A1(SLn)=K2M for n 3 n ≥ 3 n >= 3n \geq 3n≥3 and properties of the A 1 A 1 A^(1)\mathbb{A}^{1}A1-tensor product [17, LEMMA 5.1.8].
Combining the above discussion with that of the previous section, we see that for n 3 n ≥ 3 n >= 3n \geq 3n≥3, we may consider the composite maps K n + 2 M π n + 1 A 1 ( P 1 n ) G W n + 1 n K n + 2 M → Ï€ n + 1 A 1 P 1 ∧ n → G W n + 1 n K_(n+2)^(M)rarrpi_(n+1)^(A^(1))(P^(1^^n))rarrGW_(n+1)^(n)\mathbf{K}_{n+2}^{M} \rightarrow \pi_{n+1}^{\mathbb{A}^{1}}\left(\mathbb{P}^{1 \wedge n}\right) \rightarrow \mathbf{G W}_{n+1}^{n}Kn+2M→πn+1A1(P1∧n)→GWn+1n; this composite is known to be zero, but the map induced by the J-homomorphism fails to be injective. Instead, it factors through a morphism
K n + 2 M / 24 π n + 1 A 1 ( P 1 n ) G W n + 1 n K n + 2 M / 24 → Ï€ n + 1 A 1 P 1 ∧ n → G W n + 1 n K_(n+2)^(M)//24 rarrpi_(n+1)^(A^(1))(P^(1^(^^n)))rarrGW_(n+1)^(n)\mathbf{K}_{n+2}^{M} / 24 \rightarrow \pi_{n+1}^{\mathbb{A}^{1}}\left(\mathbb{P}^{1^{\wedge n}}\right) \rightarrow \mathbf{G} \mathbf{W}_{n+1}^{n}Kn+2M/24→πn+1A1(P1∧n)→GWn+1n
Furthermore, the map on the right fails to be surjective. The unstable description above is not present in the literature, but it is equivalent to the results stated in [12]. In [49], the stable motivic 1-stem was computed in the terms above: the above sequence is exact on the left stably. The next result compares the unstable group to the corresponding stable group.
Theorem 4.9. For any integer n 3 n ≥ 3 n >= 3n \geq 3n≥3, the kernel U n + 1 U n + 1 U_(n+1)\mathbf{U}_{n+1}Un+1 of the stabilization map
π n + 1 A 1 ( P 1 n ) π n + 1 A 1 ( Ω P 1 Σ P 1 P 1 n ) Ï€ n + 1 A 1 P 1 ∧ n → Ï€ n + 1 A 1 Ω P 1 ∞ Σ P 1 ∞ P 1 ∧ n pi_(n+1)^(A^(1))(P^(1^(^^n)))rarrpi_(n+1)^(A^(1))(Omega_(P^(1))^(oo)Sigma_(P^(1))^(oo)P^(1^^n))\pi_{n+1}^{\mathbb{A}^{1}}\left(\mathbb{P}^{1^{\wedge n}}\right) \rightarrow \pi_{n+1}^{\mathbb{A}^{1}}\left(\Omega_{\mathbb{P}^{1}}^{\infty} \Sigma_{\mathbb{P}^{1}}^{\infty} \mathbb{P}^{1 \wedge n}\right)Ï€n+1A1(P1∧n)→πn+1A1(ΩP1∞ΣP1∞P1∧n)
is a direct summand; the stabilization map is an isomorphism if n = 3 n = 3 n=3n=3n=3, i.e., U 4 = 0 U 4 = 0 U_(4)=0\mathbf{U}_{4}=0U4=0.
Conjecture 4.10. For n 4 n ≥ 4 n >= 4n \geq 4n≥4, the sheaf U n + 1 U n + 1 U_(n+1)\mathbf{U}_{n+1}Un+1 is zero.
Remark 4.11. Conjecture 4.10 would follow from a suitable version of the Freudenthal suspension theorem for P 1 P 1 P^(1)\mathbb{P}^{1}P1-suspension.

4.4. Splitting in corank 1

Using the results above, we can analyze the splitting problem for vector bundles in corank 1. The expected result was posed as a question by Murthy [43, P. 173] which we stated in the introduction as Conjecture 1.2. Murthy's conjecture is trivial if d = 2 d = 2 d=2d=2d=2. In [4] and [5] we established the following result, which reduces Murthy's question to Conjecture 4.10.
Theorem 4.12. Let X X XXX be a smooth affine scheme of dimension d 2 d ≥ 2 d >= 2d \geq 2d≥2 over an algebraically closed field k k kkk. A rank d 1 d − 1 d-1d-1d−1 vector bundle & & &\&& on X X XXX splits off a trivial rank 1 summand if and
only if c d 1 ( E ) C H d 1 ( X ) c d − 1 ( E ) ∈ C H d − 1 ( X ) c_(d-1)(E)inCH^(d-1)(X)c_{d-1}(\mathcal{E}) \in \mathrm{CH}^{d-1}(X)cd−1(E)∈CHd−1(X) is trivial and a secondary obstruction
o 2 ( E ) H N i s d ( X , π d 1 A 1 ( A d 1 0 ) ) o 2 ( E ) ∈ H N i s d X , Ï€ d − 1 A 1 A d − 1 ∖ 0 o_(2)(E)inH_(Nis)^(d)(X,pi_(d-1)^(A^(1))(A^(d-1)\\0))o_{2}(\mathcal{E}) \in \mathrm{H}_{\mathrm{Nis}}^{d}\left(X, \pi_{d-1}^{\mathbb{A}^{1}}\left(\mathbb{A}^{d-1} \backslash 0\right)\right)o2(E)∈HNisd(X,Ï€d−1A1(Ad−1∖0))
vanishes. This secondary obstruction vanishes if d = 3 , 4 d = 3 , 4 d=3,4d=3,4d=3,4 or if Conjecture 4.10 has a positive answer.
To establish this result, one uses the assumptions that X X XXX is smooth affine of Krull dimension d d ddd and k k kkk is algebraically closed in a strong way. Indeed, these assertions can be leveraged to show that the primary obstruction, which is a priori an Euler class, actually coincides with the ( d 1 ) ( d − 1 ) (d-1)(d-1)(d−1) st Chern class. The secondary obstruction can be described by Theorem 4.9 and the form of the secondary obstruction is extremely similar to Liao's description in Section 2.2: it is a coset in C h d ( X ) / ( S q 2 + c 1 ( E ) ) C h d 1 ( X ) C h d ( X ) / S q 2 + c 1 ( E ) ∪ C h d − 1 ( X ) Ch^(d)(X)//(Sq^(2)+c_(1)(E)uu)Ch^(d-1)(X)\mathrm{Ch}^{d}(X) /\left(\mathrm{Sq}^{2}+c_{1}(\mathcal{E}) \cup\right) \mathrm{Ch}^{d-1}(X)Chd(X)/(Sq2+c1(E)∪)Chd−1(X) where C h i ( X ) = C H i ( X ) / 2 C h i ( X ) = C H i ( X ) / 2 Ch^(i)(X)=CH^(i)(X)//2\mathrm{Ch}^{i}(X)=\mathrm{CH}^{i}(X) / 2Chi(X)=CHi(X)/2. Once more, the assumptions on X X XXX guarantee that C h d ( X ) C h d ( X ) Ch^(d)(X)\mathrm{Ch}^{d}(X)Chd(X) is trivial and thus the secondary obstruction is so as well.

4.5. The enumeration problem

If a vector bundle E E EEE splits off a free rank 1 summand, then another natural question is to enumerate the possible E E ′ E^(')E^{\prime}E′ that become isomorphic to E E EEE after adding a free rank 1 summand. This problem may also be analyzed in homotopy theoretic terms as it amounts to enumerating the number of distinct lifts. This kind of problem was studied in detail in topology by James and Thomas [33] and the same kind of analysis can be pursued in algebraic geometry.
The history of the enumeration problem in algebraic geometry goes back to early days of algebraic K-theory. Indeed, the Bass-Schanuel cancellation theorem [22] solves the enumeration problem for bundles of negative corank. Suslin's celebrated cancellation theorem [58] solved the enumeration problem in corank 0 . In all of these statements, "cancellation" means that there is a unique lift. On the other hand, Mohan Kumar observed [38] that for bundles of corank 2 , uniqueness was no longer true in general. Nevertheless, Suslin conjectured that the enumeration problem had a particularly nice solution in corank 1.
Conjecture 4.13 (Suslin's cancellation conjecture). If k k kkk is an algebraically closed field, and X X XXX is a smooth affine k k kkk-scheme of dimension d 2 d ≥ 2 d >= 2d \geq 2d≥2. If E E E\mathcal{E}E and E E ′ E^(')\mathcal{E}^{\prime}E′ are corank 1 bundles that become isomorphic after addition of a trivial rank 1 summand, then E E E\mathcal{E}E and E E ′ E^(')\mathcal{E}^{\prime}E′ are isomorphic.
The above conjecture is trivial when d = 2 d = 2 d=2d=2d=2. It was established for E E E\mathcal{E}E the trivial bundle of rank d 1 d − 1 d-1d-1d−1 in [30] and d = dim ( X ) d = dim ⁡ ( X ) d=dim(X)d=\operatorname{dim}(X)d=dim⁡(X) under the condition that ( d 1 ) ( d − 1 ) (d-1)(d-1)(d−1) ! is invertible in k k kkk. The above conjecture was also established for d = 3 d = 3 d=3d=3d=3 in [4] (assuming 2 is invertible in k k kkk ). Paralleling the results of James-Thomas in topology [33], P. Du was able to prove in [27] that Suslin's question has a positive answer for oriented vector bundles in case the cohomology group H N i s d ( X , π d A 1 ( A d 0 ) ) H N i s d X , Ï€ d A 1 A d ∖ 0 H_(Nis)^(d)(X,pi_(d)^(A^(1))(A^(d)\\0))\mathrm{H}_{\mathrm{Nis}}^{d}\left(X, \pi_{d}^{\mathbb{A}^{1}}\left(\mathbb{A}^{d} \backslash 0\right)\right)HNisd(X,Ï€dA1(Ad∖0)) vanishes. This vanishing statement would follow immediately from Conjecture 4.10.

5. VECTOR BUNDLES: NONAFFINE VARIETIES AND ALGEBRAIZABILITY

In this final section, we survey some joint work with M. J. Hopkins related to the classification of motivic vector bundles (see Definition 3.19), its relationship to the algebraizability question (see Question 3.20), and investigate the extent to which A 1 A 1 A^(1)\mathbb{A}^{1}A1-homotopy theory can be used to analyze vector bundles on projective varieties.

5.1. Descent along a Jouanolou device

If X X XXX is a smooth algebraic k k kkk-variety, then there is always the map
(5.1) Vect r ( X ) [ X , B G L r ] A 1 (5.1) Vect r ⁡ ( X ) → X , B G L r A 1 {:(5.1)Vect_(r)(X)rarr[X,BGL_(r)]_(A^(1)):}\begin{equation*} \operatorname{Vect}_{r}(X) \rightarrow\left[X, \mathrm{BGL}_{r}\right]_{\mathbb{A}^{1}} \tag{5.1} \end{equation*}(5.1)Vectr⁡(X)→[X,BGLr]A1
from rank r r rrr vector bundles to rank r r rrr motivic vector bundles. When X X XXX is affine, Theorem 3.11 guarantees that this map is a bijection, and examples show that the map fails to be an isomorphism outside of this case. Nevertheless, it is very interesting to try to quantify the failure of the above map to be a bijection.
If π : X ~ X Ï€ : X ~ → X pi: tilde(X)rarr X\pi: \tilde{X} \rightarrow XÏ€:X~→X is a Jouanolou device for X X XXX, then it follows from the definitions that the map (5.1) coincides with π : Vect r ( X ) Vect r ( X ~ ) Ï€ ∗ : Vect r ⁡ ( X ) → Vect r ⁡ ( X ~ ) pi^(**):Vect_(r)(X)rarrVect_(r)( tilde(X))\pi^{*}: \operatorname{Vect}_{r}(X) \rightarrow \operatorname{Vect}_{r}(\tilde{X})π∗:Vectr⁡(X)→Vectr⁡(X~) under the bijection of Theorem 3.11. The morphism π Ï€ pi\piÏ€ is faithfully flat by construction, and therefore, vector bundles on X X XXX are precisely vector bundles on X ~ X ~ tilde(X)\tilde{X}X~ equipped with a descent datum along π Ï€ pi\piÏ€.
Since π : X ~ X Ï€ : X ~ → X pi: tilde(X)rarr X\pi: \tilde{X} \rightarrow XÏ€:X~→X is an affine morphism, it follows that X ~ × X X ~ X ~ × X X ~ tilde(X)xx_(X) tilde(X)\tilde{X} \times_{X} \tilde{X}X~×XX~ is itself an affine scheme, and the two projections p 1 , p 2 : X ~ × X X ~ X ~ p 1 , p 2 : X ~ × X X ~ → X ~ p_(1),p_(2): tilde(X)xx_(X) tilde(X)rarr tilde(X)p_{1}, p_{2}: \tilde{X} \times_{X} \tilde{X} \rightarrow \tilde{X}p1,p2:X~×XX~→X~ are A 1 A 1 A^(1)\mathbb{A}^{1}A1-weak equivalences. Thus, pullbacks p 1 p 1 ∗ p_(1)^(**)p_{1}^{*}p1∗ and p 2 p 2 ∗ p_(2)^(**)p_{2}^{*}p2∗ are bijections on sets of isomorphism classes of vector bundles. In fact, since the relative diagonal map splits the two projections, the two pullbacks actually coincide on isomorphism classes. In descent-theoretic terms, these observations mean that any vector bundle E E E\mathscr{E}E on X ~ X ~ tilde(X)\tilde{X}X~ can always be equipped with an isomorphism p 1 E p 2 E p 1 ∗ E → ∼ p 2 ∗ E p_(1)^(**)Erarr"∼"p_(2)^(**)Ep_{1}^{*} \mathscr{E} \xrightarrow{\sim} p_{2}^{*} \mathscr{E}p1∗E→∼p2∗E, i.e., a predescent datum. Thus, the only obstruction to descending a vector bundle along π Ï€ pi\piÏ€ is whether one may choose a predescent datum that actually satisfies the cocycle condition. With this observation in mind, it seems natural to analyze the question of whether every vector bundle can be equipped with a descent datum along π Ï€ pi\piÏ€.
Question 5.1. If X X XXX is a smooth k k kkk-variety and π : X ~ X Ï€ : X ~ → X pi: tilde(X)rarr X\pi: \tilde{X} \rightarrow XÏ€:X~→X is a Jouanolou device for X X XXX, then is the pull-back map
p : Vect n ( X ) Vect n ( X ~ ) p ∗ : Vect n ⁡ ( X ) → Vect n ⁡ ( X ~ ) p^(**):Vect_(n)(X)rarrVect_(n)( tilde(X))p^{*}: \operatorname{Vect}_{n}(X) \rightarrow \operatorname{Vect}_{n}(\tilde{X})p∗:Vectn⁡(X)→Vectn⁡(X~)
surjective?
Theorem 5.2 (Asok, Fasel, Hopkins). Suppose X X XXX is a smooth projective k k kkk-variety of dimension d. If either (i) d 2 d ≤ 2 d <= 2d \leq 2d≤2 or (ii) k k kkk is algebraically closed and d 3 d ≤ 3 d <= 3d \leq 3d≤3, then Question 5.1 admits a positive answer, i.e., every vector bundle on X ~ X ~ tilde(X)\tilde{X}X~ admits a descent datum relative to π Ï€ pi\piÏ€.

5.2. Algebraizability I: obstructions

If X X XXX is a smooth complex algebraic variety, then we considered the map
Vect r ( X ) Vect r top ( X ) Vect r ⁡ ( X ) → Vect r top ⁡ ( X ) Vect_(r)(X)rarrVect_(r)^(top)(X)\operatorname{Vect}_{r}(X) \rightarrow \operatorname{Vect}_{r}^{\operatorname{top}}(X)Vectr⁡(X)→Vectrtop⁡(X)
and posed the question of characterizing its image. We observed that this map factors through the set of motivic vector bundles, so one necessary condition for a topological vector bundle to be algebraizable is that it admits a motivic lift. In particular, this means that the Chern classes of the topological vector bundle in integral cohomology must lie in the image of the cycle class map. It is natural to ask if algebraizability of Chern classes is sufficient to guarantee that a vector bundle admits a motivic left.
In case X X XXX is projective, this question has been for instance studied in [53] where it is proved that any vector bundle with algebraic Chern classes is algebraizable if dim ( X ) = 2 dim ⁡ ( X ) = 2 dim(X)=2\operatorname{dim}(X)=2dim⁡(X)=2. In case of projective threefolds, positive results are given by Atiyah-Rees and Bănică-Putinar respectively in [18] and [19]. If X X XXX is affine, the works of Swan-Murthy [44] and MurthyKumar [36] show that the answer to the question is positive if X X XXX is of dimension 3 ≤ 3 <= 3\leq 3≤3 as a consequence of the following statement: Given any pair ( α 1 , α 2 ) C H 1 ( X ) × C H 2 ( X ) α 1 , α 2 ∈ C H 1 ( X ) × C H 2 ( X ) (alpha_(1),alpha_(2))inCH^(1)(X)xxCH^(2)(X)\left(\alpha_{1}, \alpha_{2}\right) \in \mathrm{CH}^{1}(X) \times \mathrm{CH}^{2}(X)(α1,α2)∈CH1(X)×CH2(X), there exists a vector bundle E E E\mathcal{E}E on X X XXX with c i ( E ) = α i c i ( E ) = α i c_(i)(E)=alpha_(i)c_{i}(\mathscr{E})=\alpha_{i}ci(E)=αi. However, in dimension 4 , additional restrictions on Chern classes arise from the action of the motivic Steenrod algebra.
Theorem 5.3 ([10, THEOREM 2]). If X X XXX is a smooth affine 4-fold, then a pair ( c 1 , c 2 ) c 1 , c 2 ∈ (c_(1),c_(2))in\left(c_{1}, c_{2}\right) \in(c1,c2)∈ C H 1 ( X ) × C H 2 ( X ) C H 1 ( X ) × C H 2 ( X ) CH^(1)(X)xxCH^(2)(X)\mathrm{CH}^{1}(X) \times \mathrm{CH}^{2}(X)CH1(X)×CH2(X) are Chern classes of a rank 2 bundle on X X XXX if and only if c 1 , c 2 c 1 , c 2 c_(1),c_(2)c_{1}, c_{2}c1,c2 satisfy the additional condition S q 2 ( c 2 ) + c 1 c 2 = 0 S q 2 c 2 + c 1 c 2 = 0 Sq^(2)(c_(2))+c_(1)c_(2)=0\mathrm{Sq}^{2}\left(c_{2}\right)+c_{1} c_{2}=0Sq2(c2)+c1c2=0, where
S q 2 : C H 2 ( X ) C H 3 ( X ) / 2 S q 2 : C H 2 ( X ) → C H 3 ( X ) / 2 Sq^(2):CH^(2)(X)rarrCH^(3)(X)//2\mathrm{Sq}^{2}: \mathrm{CH}^{2}(X) \rightarrow \mathrm{CH}^{3}(X) / 2Sq2:CH2(X)→CH3(X)/2
is the Steenrod squaring operation, and c 1 c 2 c 1 c 2 c_(1)c_(2)c_{1} c_{2}c1c2 is the reduction modulo 2 of the cup product.
Remark 5.4. This obstruction is sufficient to identify topological vector bundles on a smooth affine fourfold X X XXX having algebraic Chern classes which are not algebraizable [10, coRolLARY 3.1.5]. One example of such an X X XXX is provided by the open complement in P 1 × P 3 P 1 × P 3 P^(1)xxP^(3)\mathbb{P}^{1} \times \mathbb{P}^{3}P1×P3 of a suitable smooth hypersurface Z Z ZZZ of bidegree (3, 4).

5.3. Algebraizability II: building motivic vector bundles

The notion of a cellular space goes back to the work of Dror Farjoun. By a cellular motivic space, we will mean a space that can be built out of the motivic spheres S p , q S p , q S^(p,q)S^{p, q}Sp,q by formation of homotopy colimits. It is straightforward to see inductively that P n P n P^(n)\mathbb{P}^{n}Pn is cellular. In the presence of cellularity assumptions, many obstructions to producing a motivic lift of a vector bundle vanish and this motivates the following conjecture.
Conjecture 5.5. If X X XXX is a smooth cellular C C C\mathbb{C}C-variety, then the map
[ X , Gr r ] A 1 Vect t o p ( X ) X , Gr r A 1 → Vect t o p ⁡ ( X ) [X,Gr_(r)]_(A^(1))rarrVect^(top)(X)\left[X, \operatorname{Gr}_{r}\right]_{\mathbb{A}^{1}} \rightarrow \operatorname{Vect}^{\mathrm{top}}(X)[X,Grr]A1→Vecttop⁡(X)
is surjective (resp. bijective).
Remark 5.6. The conjecture holds for P n P n P^(n)\mathbb{P}^{n}Pn for n 3 n ≤ 3 n <= 3n \leq 3n≤3 (this follows, for example, from the results of Schwarzenberger and Atiyah-Rees mentioned above); in these cases, bijectivity holds. For P 4 P 4 P^(4)\mathbb{P}^{4}P4, the "surjective" formulation of Conjecture 5.5 is known, but the "bijective" formulation is not.
We now analyze Conjecture 5.5 for a class of "interesting" topological vector bundles on P n P n P^(n)\mathbb{P}^{n}Pn introduced by E. Rees and L. Smith. We briefly recall the construction of these topological vector bundles here. By a classical result of Serre [54, PROPOsition 11], we know that if p p ppp is a prime, then the p p ppp-primary component of π 4 p 3 ( S 3 ) Ï€ 4 p − 3 S 3 pi_(4p-3)(S^(3))\pi_{4 p-3}\left(S^{3}\right)Ï€4p−3(S3) is isomorphic to Z / p Z / p Z//p\mathbb{Z} / pZ/p, generated by the composite of a generator α 1 α 1 alpha_(1)\alpha_{1}α1 of the p p ppp-primary component of π 2 p ( S 3 ) Ï€ 2 p S 3 pi_(2p)(S^(3))\pi_{2 p}\left(S^{3}\right)Ï€2p(S3) and the ( 2 p 3 ) ( 2 p − 3 ) (2p-3)(2 p-3)(2p−3) rd suspension of itself; we will write α 1 2 α 1 2 alpha_(1)^(2)\alpha_{1}^{2}α12 for this class.
The map P n S 2 n P n → S 2 n P^(n)rarrS^(2n)\mathbb{P}^{n} \rightarrow S^{2 n}Pn→S2n that collapses P n 1 P n − 1 P^(n-1)\mathbb{P}^{n-1}Pn−1 to a point determines a function
[ S 2 n 1 , S 3 ] [ S 2 n , BSU ( 2 ) ] [ P n , BSU ( 2 ) ] S 2 n − 1 , S 3 ≅ S 2 n , BSU ⁡ ( 2 ) → P n , BSU ⁡ ( 2 ) [S^(2n-1),S^(3)]~=[S^(2n),BSU(2)]rarr[P^(n),BSU(2)]\left[S^{2 n-1}, S^{3}\right] \cong\left[S^{2 n}, \operatorname{BSU}(2)\right] \rightarrow\left[\mathbb{P}^{n}, \operatorname{BSU}(2)\right][S2n−1,S3]≅[S2n,BSU⁡(2)]→[Pn,BSU⁡(2)]
Rees established that the class α 1 2 α 1 2 alpha_(1)^(2)\alpha_{1}^{2}α12 determines a nontrivial rank 2 vector bundle ξ p ξ p ∈ xi_(p)in\xi_{p} \inξp∈ [ P 2 p 1 , B S U ( 2 ) ] P 2 p − 1 , B S U ( 2 ) [P^(2p-1),BSU(2)]\left[\mathbb{P}^{2 p-1}, \mathrm{BSU}(2)\right][P2p−1,BSU(2)]; we will refer to this bundle as a Rees bundle [48]. By construction, ξ p ξ p xi_(p)\xi_{p}ξp is a nontrivial rank 2 bundle with trivial Chern classes.
The motivation for Rees' construction originated from results of Grauert-Schneider [32]. If the bundles ξ p ξ p xi_(p)\xi_{p}ξp were algebraizable, then the fact that they have trivial Chern classes would imply they were necessarily unstable by Barth's results on Chern classes of stable vector bundles [21, CORoLLARY 1 P. 127] (here, stability means slope stability in the sense of Mumford). Grauert and Schneider analyzed unstable rank 2 vector bundles on projective space and they aimed to prove that such vector bundles were necessarily direct sums of line bundles; this assertion is now sometimes known as the Grauert-Schneider conjecture. In view of the Grauert-Schneider conjecture, the bundles ξ p ξ p xi_(p)\xi_{p}ξp should not be algebraizable. On the other hand, one of the motivations for Conjecture 5.5 is the following result.
Theorem 5.7 ([11, THEOREM 2.2.16]). For every prime number p p ppp, the bundle ξ p ξ p xi_(p)\xi_{p}ξp lifts to a class in [ P 2 p 1 , G r 2 ] A 1 P 2 p − 1 , G r 2 A 1 [P^(2p-1),Gr_(2)]_(A^(1))\left[\mathbb{P}^{2 p-1}, \mathrm{Gr}_{2}\right]_{\mathbb{A}^{1}}[P2p−1,Gr2]A1.
Remark 5.8. This is established by constructing motivic homotopy classes lifting α 1 α 1 alpha_(1)\alpha_{1}α1 and α 1 2 α 1 2 alpha_(1)^(2)\alpha_{1}^{2}α12. In our situation, the collapse map takes the form
P n S n , n P n → S n , n P^(n)rarrS^(n,n)\mathbb{P}^{n} \rightarrow S^{n, n}Pn→Sn,n
and the lift must come from an element of [ S n 1 , n , S L 2 ] A 1 S n − 1 , n , S L 2 A 1 [S^(n-1,n),SL_(2)]_(A^(1))\left[S^{n-1, n}, S L_{2}\right]_{\mathbb{A}^{1}}[Sn−1,n,SL2]A1. The class α 1 α 1 alpha_(1)\alpha_{1}α1 can be lifted using ideas related to those discussed in 4.1 in conjunction with a motivic version of Serre's classical p p ppp-local splitting of compact Lie groups [9, THEOREM 2], the resulting lift has the wrong weight to lift to a group as above. Since the class α 1 2 α 1 2 alpha_(1)^(2)\alpha_{1}^{2}α12 is torsion, we can employ a weightshifting mechanism to fix this issue. In this direction, there are host of other vector bundles that are analogous to the Rees bundles that one might investigate from this point of view, e.g., bundles that can be built out of Toda's unstable α α alpha\alphaα-family [62]. Likewise, even the surjectivity assertion in Conjecture 5.5 is unknown for P 5 P 5 P^(5)\mathbb{P}^{5}P5.
5.9 (The Wilson Space Hypothesis). To close, we briefly sketch an approach to the resolution of Conjecture 5.5 building on Mike Hopkins' Wilson Space Hypothesis. The latter asserts that the Voevodsky motive of the P 1 P 1 P^(1)\mathbb{P}^{1}P1-infinite loop spaces Ω P 1 Σ P 1 n Ω P 1 ∞ Σ P 1 n Omega_(P^(1))^(oo)Sigma_(P^(1))^(n)\Omega_{\mathbb{P}^{1}}^{\infty} \Sigma_{\mathbb{P}^{1}}^{n}ΩP1∞ΣP1n MGL arising from algebraic cobordism are pure Tate (the space is "homologically even"); this hypothesis is an algebro-geometric version of a result of Steve Wilson on the infinite loop spaces of the classical cobordism spectrum.
The motivic version of the unstable Adams-Novikov resolution for B G L r B G L r BGL_(r)\mathrm{BGL}_{r}BGLr yields a spectral sequence that, under the cellularity assumption on X X XXX should converge to a (completion of) the set of rank r r rrr motivic vector bundles on X X XXX. The resulting spectral sequence can be compared to its topological counterpart and Wilson Space Hypothesis combined with the cellularity assumption on X X XXX would imply that the two spectral sequences coincide.

ACKNOWLEDGMENTS

The authors would like to acknowledge the profound influence of M. Hopkins and F. Morel on their work. Many of the results presented here would not have been possible without their insights and vision. Both authors would also like to thank all their colleagues, coauthors, and friends for their help and support.

FUNDING

Aravind Asok was partially supported by NSF Awards DMS-1802060 and DMS-2101898.

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ARAVIND ASOK

Department of Mathematics, University of Southern California, 3620 S. Vermont Ave., Los Angeles, CA 90089-2532, USA, asok @usc.edu

JEAN FASEL

Institut Fourier, Université Grenoble-Alpes, CS 40700, 38058 Grenoble cedex 9, France, jean.fasel@univ-grenoble-alpes.fr

THE UNREASONABLE EFFECTIVENESS OF WALL-CROSSING IN ALGEBRAIC GEOMETRY AREND BAYER AND EMANUELE MACRÌ

ABSTRACT

We survey applications of Bridgeland stability conditions in algebraic geometry and discuss open questions for future research.

MATHEMATICS SUBJECT CLASSIFICATION 2020

Primary 14F08; Secondary 14F17, 14H51, 14J28, 14J32, 14J42

KEYWORDS

Bridgeland stability conditions, K3 categories, hyperkähler varieties, Calabi-Yau threefolds, Brill-Noether theory, cubic fourfolds, Donaldson-Thomas theory

1. INTRODUCTION

Bridgeland stability conditions and wall-crossing have provided answers to many questions in algebraic geometry a priori unrelated to derived categories, including hyperkähler varieties-their rational curves, their birational geometry, their automorphisms, and their moduli spaces-, Brill-Noether questions, Noether-Lefschetz loci, geometry of cubic fourfolds, or higher-rank Donaldson-Thomas theory. Our goal is to answer the question: why? What makes these techniques so effective, and what exactly do they add beyond, for example, classical vector bundle techniques?
The common underlying strategy can be roughly summarized as follows. For each stability condition on a derived category D b ( X ) D b ( X ) D^(b)(X)\mathrm{D}^{\mathrm{b}}(X)Db(X) of an algebraic variety X X XXX and each numerical class, moduli spaces of semistable objects in D b ( X ) D b ( X ) D^(b)(X)\mathrm{D}^{\mathrm{b}}(X)Db(X) exist as proper algebraic spaces. This formalism includes many previously studied moduli spaces: moduli spaces of Gieseker- or slope-stable sheaves, of stable pairs, or of certain equivalences classes of rational curves in X X XXX. The set of stability conditions on D b ( X ) D b ( X ) D^(b)(X)\mathrm{D}^{\mathrm{b}}(X)Db(X) has the structure of a complex manifold; when we vary the stability condition, stability of a given object only changes when we cross the walls of a well-defined wall and chamber structure.
The typical ingredients when approaching a problem with stability conditions are the following:
(large volume) There is a point in the space of stability conditions where stable objects have a "classical" interpretation, e.g. as Gieseker-stable sheaves.
(point of interest) There is a point in the space of stability conditions where stability has strong implications, e.g., vanishing properties, or even there is no semistable object of a given numerical class.
(wall-crossing) It is possible to analyze the finite set of walls between these two points, and how stability changes when crossing each wall.
In general, it is quite clear from the problem which are the points of interest, and the main difficulty consists in analyzing the walls. In the ideal situation, which leads to sharp exact results, these walls can be characterized purely numerically; there are only a few such ideal situations, K3 surfaces being one of them. Otherwise, the study of walls can get quite involved, even though there are now a number of more general results available, e.g., a wallcrossing formula for counting invariants arising from moduli spaces.
We illustrate the case of K3 surfaces, or more generally K3 categories, in Section 3, with applications to hyperkähler varieties, to Brill-Noether theory of curves, and to the geometry of special cubic fourfolds. The study of other surfaces or the higher-dimensional case becomes more technical, and the existence of Bridgeland stability conditions is not yet known in full generality. There are weaker notions of stability, which in the threefold case already lead to striking results. We give an overview of this circle of ideas in Section 4, along with three applications related to curves. We give a brief introduction to stability conditions in Section 2, and pose some questions for future research in Section 5.
Derived categories of coherent sheaves on varieties have been hugely influential in recent years; we refer to [ 17 , 19 , 46 , 74 ] [ 17 , 19 , 46 , 74 ] [17,19,46,74][17,19,46,74][17,19,46,74] for an overview of the theory. Moduli spaces of sheaves on K3 surfaces have largely been influenced by [61]; we refer to [63] for an overview of applications of these techniques and to [39] for the higher-dimensional case of hyperkähler manifolds. For the original motivation from physics, we refer to [26,41].
Our survey completely omits the quickly developing theory and applications of stability conditions on Kuznetsov components of Fano threefolds. We also will not touch the rich subject of extra structures on spaces of stability conditions, developed, for example, in the foundational papers [22,23]; we also refer to [72] for a symplectic perspective.

2. STABILITY CONDITIONS ON DERIVED CATEGORIES

Recall slope-stability for sheaves on an integral projective curve C C CCC : we set
μ ( E ) = deg E rk E ( , + ] , visualized by Z ( E ) = deg E + i r k E μ ( E ) = deg ⁡ E rk ⁡ E ∈ ( − ∞ , + ∞ ] ,  visualized by  Z ( E ) = − deg ⁡ E + i r k E mu(E)=(deg E)/(rk E)in(-oo,+oo],quad" visualized by "Z(E)=-deg E+irkE\mu(E)=\frac{\operatorname{deg} E}{\operatorname{rk} E} \in(-\infty,+\infty], \quad \text { visualized by } Z(E)=-\operatorname{deg} E+i \mathrm{rk} Eμ(E)=deg⁡Erk⁡E∈(−∞,+∞], visualized by Z(E)=−deg⁡E+irkE
and call a sheaf slope-semistable if every subsheaf F E F ⊂ E F sub EF \subset EF⊂E satisfies μ ( F ) μ ( F ) μ ( F ) ≤ μ ( F ) mu(F) <= mu(F)\mu(F) \leq \mu(F)μ(F)≤μ(F). The set of semistable sheaves of fixed rank and degree is bounded and can be parameterized by a projective moduli space. Moreover, semistable sheaves generate Coh ( C ) Coh ⁡ ( C ) Coh(C)\operatorname{Coh}(C)Coh⁡(C), in the sense that every sheaf E E EEE admits a (unique and functorial) Harder-Narasimhan (HN) filtration
0 = E 0 E 1 E 2 E m = E 0 = E 0 ⊂ E 1 ⊂ E 2 ⊂ ⋯ ⊂ E m = E 0=E_(0)subE_(1)subE_(2)sub cdots subE_(m)=E0=E_{0} \subset E_{1} \subset E_{2} \subset \cdots \subset E_{m}=E0=E0⊂E1⊂E2⊂⋯⊂Em=E
with E l / E l 1 E l / E l − 1 E_(l)//E_(l-1)E_{l} / E_{l-1}El/El−1 semistable, and μ ( E 1 ) > μ ( E 2 / E 1 ) > > μ ( E m / E m 1 ) μ E 1 > μ E 2 / E 1 > ⋯ > μ E m / E m − 1 mu(E_(1)) > mu(E_(2)//E_(1)) > cdots > mu(E_(m)//E_(m-1))\mu\left(E_{1}\right)>\mu\left(E_{2} / E_{1}\right)>\cdots>\mu\left(E_{m} / E_{m-1}\right)μ(E1)>μ(E2/E1)>⋯>μ(Em/Em−1).
How to generalize this to a variety X X XXX of dimension n 2 n ≥ 2 n >= 2n \geq 2n≥2 ? Given a polarization H H HHH, one can define the slope μ H μ H mu_(H)\mu_{H}μH using H n 1 ch 1 ( E ) H n − 1 â‹… ch 1 ⁡ ( E ) H^(n-1)*ch_(1)(E)H^{n-1} \cdot \operatorname{ch}_{1}(E)Hn−1â‹…ch1⁡(E) as the degree. To distinguish, e.g., the slope of the structure sheaf O X O X O_(X)\mathcal{O}_{X}OX from that of an ideal sheaf I x O X I x ⊂ O X I_(x)subO_(X)\mathscr{I}_{x} \subset \mathcal{O}_{X}Ix⊂OX for x X x ∈ X x in Xx \in Xx∈X, we can further refine the notion of slope and use lower-degree terms of the Hilbert polynomial p E ( m ) = χ ( E ( m H ) ) p E ( m ) = χ ( E ( m H ) ) p_(E)(m)=chi(E(mH))p_{E}(m)=\chi(E(m H))pE(m)=χ(E(mH)) as successive tie breakers; this yields Gieseker stability.
One of the key insights in Bridgeland's notion of stability conditions introduced in [20] is that instead we can, in fact, still use a notion of slope-stability, defined as the quotient of "degree" by "rank." The price we have to pay is to replace Coh ( X ) Coh ⁡ ( X ) Coh(X)\operatorname{Coh}(X)Coh⁡(X) by another abelian subcategory A A A\mathcal{A}A of the bounded derived category D b ( X ) D b ( X ) D^(b)(X)\mathrm{D}^{\mathrm{b}}(X)Db(X) of coherent sheaves on X X XXX, and to generalize the notions of "degree" and "rank" (combined into a central charge Z Z ZZZ as above).
To motivate the definition, consider again slope-stability for a curve C C CCC. First, for ϕ ( 0 , 1 ] Ï• ∈ ( 0 , 1 ] phi in(0,1]\phi \in(0,1]ϕ∈(0,1], let P ( ϕ ) Coh ( C ) D b ( C ) P ( Ï• ) ⊂ Coh ⁡ ( C ) ⊂ D b ( C ) P(phi)sub Coh(C)subD^(b)(C)\mathcal{P}(\phi) \subset \operatorname{Coh}(C) \subset \mathrm{D}^{\mathrm{b}}(C)P(Ï•)⊂Coh⁡(C)⊂Db(C) be the category of slope-semistable sheaves E E EEE with Z ( E ) R > 0 e i π ϕ Z ( E ) ∈ R > 0 â‹… e i Ï€ Ï• Z(E)inR_( > 0)*e^(i pi phi)Z(E) \in \mathbb{R}_{>0} \cdot e^{i \pi \phi}Z(E)∈R>0â‹…eiπϕ, i.e., of slope μ ( E ) = cot ( π ϕ ) μ ( E ) = − cot ⁡ ( Ï€ Ï• ) mu(E)=-cot(pi phi)\mu(E)=-\cot (\pi \phi)μ(E)=−cot⁡(πϕ), and let P ( ϕ + n ) = P ( ϕ ) [ n ] P ( Ï• + n ) = P ( Ï• ) [ n ] P(phi+n)=P(phi)[n]\mathcal{P}(\phi+n)=\mathcal{P}(\phi)[n]P(Ï•+n)=P(Ï•)[n] for n Z n ∈ Z n inZn \in \mathbb{Z}n∈Z be the set of semistable objects of phase ϕ + n Ï• + n phi+n\phi+nÏ•+n. Every complex E D b ( C ) E ∈ D b ( C ) E inD^(b)(C)E \in \mathrm{D}^{\mathrm{b}}(C)E∈Db(C) has a filtration into its cohomology objects H l ( E ) [ l ] H l ( E ) [ − l ] H^(l)(E)[-l]\mathscr{H}^{l}(E)[-l]Hl(E)[−l]. We can combine this with the classical HN filtration of H l ( E ) [ l ] H l ( E ) [ − l ] H^(l)(E)[-l]\mathscr{H}^{l}(E)[-l]Hl(E)[−l] for each l l lll to obtain a finer filtration for E E EEE where every filtration quotient is semistable, i.e., an object of P ( ϕ ) P ( Ï• ) P(phi)\mathcal{P}(\phi)P(Ï•) for ϕ R Ï• ∈ R phi inR\phi \in \mathbb{R}ϕ∈R. The properties of this structure are axiomatized by conditions (1)-(4) in Definition 2.1 below. But crucially it can always be obtained from slope-stability in an abelian category A A A\mathcal{A}A; we just have to generalize the setting A D b ( A ) A ⊂ D b ( A ) AsubD^(b)(A)\mathcal{A} \subset \mathrm{D}^{\mathrm{b}}(\mathcal{A})A⊂Db(A) to A D A ⊂ D AsubD\mathscr{A} \subset \mathscr{D}A⊂D being the "heart of a bounded t t t\mathrm{t}t-structure" in a triangulated category.
When combined with the remaining conditions in Definition 2.1, the main payoff are the strong deformation and wall-crossing properties of Bridgeland stability conditions. Given any small deformation of "rank" and "degree" (equivalently, of the central charge Z Z ZZZ ), we can accordingly adjust the abelian category A A A\mathcal{A}A (or, equivalently, the set of semistable objects P P P\mathcal{P}P ) and obtain a new stability condition. Along such a deformation, moduli spaces of semistable objects undergo very well-behaved wall-crossing transformations.

2.1. Bridgeland stability conditions

We now consider more generally an admissible subcategory D D D\mathscr{D}D of D b ( X ) D b ( X ) D^(b)(X)\mathrm{D}^{\mathrm{b}}(X)Db(X) for a smooth and proper variety X X XXX over a field k k kkk, namely a full triangulated subcategory whose inclusion D D b ( X ) D ↪ D b ( X ) D↪D^(b)(X)\mathscr{D} \hookrightarrow \mathrm{D}^{\mathrm{b}}(X)D↪Db(X) admits both a left and a right adjoint. For instance, D = D b ( X ) D = D b ( X ) D=D^(b)(X)\mathscr{D}=\mathrm{D}^{\mathrm{b}}(X)D=Db(X); otherwise we think of D D D\mathscr{D}D as a smooth and proper noncommutative variety.
We fix a finite rank free abelian group Λ Î› Lambda\LambdaΛ and a group homomorphism
v : K 0 ( D ) Λ v : K 0 ( D ) → Λ v:K_(0)(D)rarr Lambdav: K_{0}(\mathscr{D}) \rightarrow \Lambdav:K0(D)→Λ
from the Grothendieck group K 0 ( D ) K 0 ( D ) K_(0)(D)K_{0}(\mathscr{D})K0(D) of D D D\mathscr{D}D to Λ Î› Lambda\LambdaΛ.
Definition 2.1. A Bridgeland stability condition on D D D\mathscr{D}D with respect to ( v , Λ ) ( v , Λ ) (v,Lambda)(v, \Lambda)(v,Λ) is a pair σ = ( Z , P ) σ = ( Z , P ) sigma=(Z,P)\sigma=(Z, \mathcal{P})σ=(Z,P) where
  • Z : Λ C Z : Λ → C Z:Lambda rarrCZ: \Lambda \rightarrow \mathbb{C}Z:Λ→C is a group homomorphism, called central charge, and
  • P = ( P ( ϕ ) ) ϕ R P = ( P ( Ï• ) ) Ï• ∈ R P=(P(phi))_(phi inR)\mathscr{P}=(\mathscr{P}(\phi))_{\phi \in \mathbb{R}}P=(P(Ï•))ϕ∈R is a collection of full additive subcategories P ( ϕ ) D P ( Ï• ) ⊂ D P(phi)subD\mathscr{P}(\phi) \subset \mathscr{D}P(Ï•)⊂D satisfying the following conditions:
(1) for all nonzero E P ( ϕ ) E ∈ P ( Ï• ) E inP(phi)E \in \mathcal{P}(\phi)E∈P(Ï•), we have Z ( v ( E ) ) R > 0 e i π ϕ Z ( v ( E ) ) ∈ R > 0 â‹… e i Ï€ Ï• Z(v(E))inR_( > 0)*e^(i pi phi)Z(v(E)) \in \mathbb{R}_{>0} \cdot e^{i \pi \phi}Z(v(E))∈R>0â‹…eiπϕ;
(2) for all ϕ R Ï• ∈ R phi inR\phi \in \mathbb{R}ϕ∈R, we have P ( ϕ + 1 ) = P ( ϕ ) [ 1 ] P ( Ï• + 1 ) = P ( Ï• ) [ 1 ] P(phi+1)=P(phi)[1]\mathcal{P}(\phi+1)=\mathcal{P}(\phi)[1]P(Ï•+1)=P(Ï•)[1];
(3) if ϕ 1 > ϕ 2 Ï• 1 > Ï• 2 phi_(1) > phi_(2)\phi_{1}>\phi_{2}Ï•1>Ï•2 and E j P ( ϕ j ) E j ∈ P Ï• j E_(j)inP(phi_(j))E_{j} \in \mathcal{P}\left(\phi_{j}\right)Ej∈P(Ï•j), then Hom ( E 1 , E 2 ) = 0 Hom ⁡ E 1 , E 2 = 0 Hom(E_(1),E_(2))=0\operatorname{Hom}\left(E_{1}, E_{2}\right)=0Hom⁡(E1,E2)=0;
(4) (Harder-Narasimhan filtrations) for all nonzero E D E ∈ D E inDE \in \mathscr{D}E∈D, there exist real numbers ϕ 1 > ϕ 2 > > ϕ m Ï• 1 > Ï• 2 > ⋯ > Ï• m phi_(1) > phi_(2) > cdots > phi_(m)\phi_{1}>\phi_{2}>\cdots>\phi_{m}Ï•1>Ï•2>⋯>Ï•m and a finite sequence of morphisms
0 = E 0 s 1 E 1 s 2 s m E m = E 0 = E 0 → s 1 E 1 → s 2 ⋯ → s m E m = E 0=E_(0)rarr"s_(1)"E_(1)rarr"s_(2)"cdotsrarr"s_(m)"E_(m)=E0=E_{0} \xrightarrow{s_{1}} E_{1} \xrightarrow{s_{2}} \cdots \xrightarrow{s_{m}} E_{m}=E0=E0→s1E1→s2⋯→smEm=E
such that the cone of s l s l s_(l)s_{l}sl is a non-zero object of P ( ϕ l ) P Ï• l P(phi_(l))\mathcal{P}\left(\phi_{l}\right)P(Ï•l);
(5) (support property) there exists a quadratic form Q Q QQQ on Λ R = Λ R Λ R = Λ ⊗ R Lambda_(R)=Lambda oxR\Lambda_{\mathbb{R}}=\Lambda \otimes \mathbb{R}ΛR=Λ⊗R such that
  • the kernel of Z Z ZZZ is negative definite with respect to Q Q QQQ, and
  • for all E P ( ϕ ) E ∈ P ( Ï• ) E inP(phi)E \in \mathcal{P}(\phi)E∈P(Ï•) for any ϕ Ï• phi\phiÏ• we have Q ( v ( E ) ) 0 Q ( v ( E ) ) ≥ 0 Q(v(E)) >= 0Q(v(E)) \geq 0Q(v(E))≥0;
(6) (openness of stability) the property of being in P ( ϕ ) P ( Ï• ) P(phi)\mathcal{P}(\phi)P(Ï•) is open in families of objects in D D D\mathscr{D}D over any scheme;
(7) (boundedness) objects in P ( ϕ ) P ( Ï• ) P(phi)\mathcal{P}(\phi)P(Ï•) with fixed class v Λ v ∈ Λ v in Lambdav \in \Lambdav∈Λ are parameterized by a k k kkk-scheme of finite type.
An object of the subcategory P ( ϕ ) P ( Ï• ) P(phi)\mathcal{P}(\phi)P(Ï•) is called σ σ sigma\sigmaσ-semistable of phase ϕ Ï• phi\phiÏ•, and σ σ sigma\sigmaσ-stable if it admits no non-trivial subobject in P ( ϕ ) P ( Ï• ) P(phi)\mathcal{P}(\phi)P(Ï•). The set of Bridgeland stability conditions on D D D\mathscr{D}D is denoted by Stab ( D ) Stab ⁡ ( D ) Stab(D)\operatorname{Stab}(\mathscr{D})Stab⁡(D), where we omit the dependence on ( v , Λ ) ( v , Λ ) (v,Lambda)(v, \Lambda)(v,Λ) from the notation.
Conditions (1)-(4) form the original definition in [20] and ensure we have a notion of slope-stability. The support property is necessary to show that stability conditions vary continuously (see Theorem 2.2 below) and admit a well-behaved wall and chamber structure: fundamentally, this is due to the simple linear algebra consequence that given C > 0 C > 0 C > 0C>0C>0, there are only finitely many classes w Λ w ∈ Λ w in Lambdaw \in \Lambdaw∈Λ of semistable objects with | Z ( w ) | < C | Z ( w ) | < C |Z(w)| < C|Z(w)|<C|Z(w)|<C. Conditions (6) and (7) were introduced in [73,74], with similar versions appearing previously in [42, sECTION 3]; they guarantee the existence of moduli spaces of semistable objects.
Theorem 2.2 (Bridgeland deformation theorem). The set Stab ( D ) Stab ⁡ ( D ) Stab(D)\operatorname{Stab}(\mathscr{D})Stab⁡(D) has the structure of a complex manifold such that the natural map
Z : Stab ( D ) Hom ( Λ , C ) , ( Z , P ) Z Z : Stab ⁡ ( D ) → Hom ⁡ ( Λ , C ) , ( Z , P ) ↦ Z Z:Stab(D)rarr Hom(Lambda,C),quad(Z,P)|->Z\mathcal{Z}: \operatorname{Stab}(\mathscr{D}) \rightarrow \operatorname{Hom}(\Lambda, \mathbb{C}), \quad(Z, \mathcal{P}) \mapsto ZZ:Stab⁡(D)→Hom⁡(Λ,C),(Z,P)↦Z
is a local isomorphism at every point.
For conditions (1)-(5), this is a reformulation of Bridgeland's main result [20, THEOREM 1.2]. It says that σ = ( Z , P ) σ = ( Z , P ) sigma=(Z,P)\sigma=(Z, \mathcal{P})σ=(Z,P) can be deformed uniquely given a small deformation of Z Z Z ⇝ Z ′ Z⇝Z^(')Z \leadsto Z^{\prime}Z⇝Z′, roughly as long as Z ( E ) 0 Z ′ ( E ) ≠ 0 Z^(')(E)!=0Z^{\prime}(E) \neq 0Z′(E)≠0 remains true for all σ σ sigma\sigmaσ-semistable objects E E EEE. (More precisely, any path where Q Q QQQ remains negative definite on Ker Z Ker ⁡ Z ′ Ker Z^(')\operatorname{Ker} Z^{\prime}Ker⁡Z′ can be lifted uniquely to a path in Stab ( D ) Stab ⁡ ( D ) Stab(D)\operatorname{Stab}(\mathscr{D})Stab⁡(D).) With the additional conditions (6) and (7), Theorem 2.2 was proved in [73, THEOREM 3.20] and [67, SECTION 4.4], where the most difficult aspect is to show that openness of stability is preserved under deformations.
The theory has been developed over an arbitrary base scheme in [8]. A stability condition over a base is the datum of a stability condition on each fiber, such that families of objects over the base have locally constant central charges, satisfy openness of stability, and a global notion of HN filtration after base change to a one-dimensional scheme; moreover, we impose a global version of the support property and of boundedness. An analogue of Theorem 2.2 holds; differently to the absolute case, assuming the support property is not enough and the proof requires the additional conditions (6) and (7).
The construction of Bridgeland stability conditions is discussed in Section 4; in particular, they exist on surfaces and certain threefolds.

2.2. Stability conditions as polarizations

It was first suggested in the arXiv version of [21] to think of σ σ sigma\sigmaσ as a polarization of the noncommutative variety D D D\mathscr{D}D. We now review some results partly justifying this analogy. A polarization of a variety X X XXX by an ample line bundle H H HHH gives projective moduli spaces of H H HHH-Gieseker-stable sheaves; the following two results provide an analogue.
Theorem 2.3 (Toda, Alper, Halpern-Leistner, Heinloth). Given σ Stab ( D ) σ ∈ Stab ⁡ ( D ) sigma in Stab(D)\sigma \in \operatorname{Stab}(\mathscr{D})σ∈Stab⁡(D) and v Λ v ∈ Λ v in Lambdav \in \Lambdav∈Λ, there is a finite type Artin stack M σ ( v ) M σ ( v ) M_(sigma)(v)\mathcal{M}_{\sigma}(v)Mσ(v) of σ σ sigma\sigmaσ-semistable objects E E EEE with v ( E ) = v v ( E ) = v v(E)=vv(E)=vv(E)=v and fixed phase. In characteristic zero, it has a proper good moduli space M σ ( v ) M σ ( v ) M_(sigma)(v)M_{\sigma}(v)Mσ(v) in the sense of Alper.
Proof. The existence as Artin stack is [73, THEOREM 3.20], while the existence of a good moduli space is proven in [2, THEOREM 7.25]; See alSO [8, THEOREM 21.24].
Theorem 2.4. The algebraic space M σ ( v ) M σ ( v ) M_(sigma)(v)M_{\sigma}(v)Mσ(v) admits a Cartier divisor σ â„“ σ â„“_(sigma)\ell_{\sigma}ℓσ that has strictly positive degree on every curve. In characteristic zero, if M σ ( v ) M σ ( v ) M_(sigma)(v)M_{\sigma}(v)Mσ(v) is smooth, or more generally if it has Q Q Q\mathbb{Q}Q-factorial log-terminal singularities, then M σ ( v ) M σ ( v ) M_(sigma)(v)M_{\sigma}(v)Mσ(v) is projective.
Proof. The existence of the Cartier divisor and its properties is the Positivity Lemma in [12], see also [8, tHEOREM 21.25]. The projectivity follows from [76, COROLLARY 3.4].
As studied extensively in Donaldson theory in the 1990s, the Gieseker-moduli spaces may change as H H HHH crosses walls in the ample cone.
Theorem 2.5. Fix a vector v Λ v ∈ Λ v in Lambdav \in \Lambdav∈Λ. Then there exists a locally finite union W v W v W_(v)\mathcal{W}_{v}Wv of realcodimension one submanifolds in Stab ( D ) Stab ⁡ ( D ) Stab(D)\operatorname{Stab}(\mathscr{D})Stab⁡(D), called walls, such that on every connected component C C C\mathcal{C}C of the complement Stab ( D ) W v Stab ⁡ ( D ) ∖ W v Stab(D)\\W_(v)\operatorname{Stab}(\mathscr{D}) \backslash \mathcal{W}_{v}Stab⁡(D)∖Wv, called a chamber, the moduli space M σ ( v ) M σ ( v ) M_(sigma)(v)M_{\sigma}(v)Mσ(v) is independent of the choice σ σ ∈ â„“ sigma inâ„“\sigma \in \mathcal{\ell}σ∈ℓ.
Theorem 2.5 follows from the results in [21, SECTION 9]; see also [73, PROPOSITION 2.8] and [10, PROPOSITION 3.3]. The set W v W v W_(v)\mathcal{W}_{v}Wv consists of stability conditions for which there exists an exact triangle A E B A → E → B A rarr E rarr BA \rightarrow E \rightarrow BA→E→B of semistable objects of the same phase with v ( E ) = v v ( E ) = v v(E)=vv(E)=vv(E)=v, but v ( A ) v ( A ) v(A)v(A)v(A) not proportional to v v vvv. Locally, the wall is defined by Z ( A ) Z ( A ) Z(A)Z(A)Z(A) being proportional to Z ( E ) Z ( E ) Z(E)Z(E)Z(E), and the objects E E EEE is unstable on the side where arg ( Z ( A ) ) > arg ( Z ( E ) ) arg ⁡ ( Z ( A ) ) > arg ⁡ ( Z ( E ) ) arg(Z(A)) > arg(Z(E))\arg (Z(A))>\arg (Z(E))arg⁡(Z(A))>arg⁡(Z(E)); often it is stable near the wall on the other side, e.g., when A A AAA and B B BBB are stable and the extension is nontrivial. The support property (5) is again crucial in the proof of Theorem 2.5: it constrains the classes a = v ( A ) , b = v ( B ) a = v ( A ) , b = v ( B ) a=v(A),b=v(B)a=v(A), b=v(B)a=v(A),b=v(B) involved in a wall, and locally that produces a finite set.
Sometimes, one can describe W v W v W_(v)\mathcal{W}_{v}Wv completely, namely when we know which of the moduli spaces M σ ( a ) M σ ( a ) M_(sigma)(a)M_{\sigma}(a)Mσ(a) and M σ ( b ) M σ ( b ) M_(sigma)(b)M_{\sigma}(b)Mσ(b) are nonempty.

2.3. K3 categories

Such descriptions of W v W v W_(v)\mathcal{W}_{v}Wv have been particularly powerful in the case of K 3 K 3 K3\mathrm{K} 3K3 categories; it has also been carried out completely for D b ( P 2 ) D b P 2 D^(b)(P^(2))\mathrm{D}^{\mathrm{b}}\left(\mathbb{P}^{2}\right)Db(P2), where the answer is more involved [24,54]. For this section, we work over the complex numbers and let D D D\mathscr{D}D be
(1) the derived category D = D b ( S ) D = D b ( S ) D=D^(b)(S)\mathscr{D}=\mathrm{D}^{\mathrm{b}}(S)D=Db(S) of a smooth projective K 3 K 3 K3\mathrm{K} 3K3 surface, or
(2) the Kuznetsov component
D = K u ( Y ) = O Y O Y ( 1 ) O Y ( 2 ) D b ( Y ) D = K u ( Y ) = O Y ⊥ ∩ O Y ( 1 ) ⊥ ∩ O Y ( 2 ) ⊥ ⊂ D b ( Y ) D=Ku(Y)=O_(Y)^(_|_)nnO_(Y)(1)^(_|_)nnO_(Y)(2)^(_|_)subD^(b)(Y)\mathscr{D}=\mathcal{K} u(Y)=\mathcal{O}_{Y}^{\perp} \cap \mathcal{O}_{Y}(1)^{\perp} \cap \mathcal{O}_{Y}(2)^{\perp} \subset \mathrm{D}^{\mathrm{b}}(Y)D=Ku(Y)=OY⊥∩OY(1)⊥∩OY(2)⊥⊂Db(Y)
of the derived category of a smooth cubic fourfold Y Y YYY, or
(3) the Kuznetsov component of a Gushel-Mukai fourfold defined in [47].
In (1) we can also allow a Brauer twist; one expects further examples of Kuznetsov components of Fano varieties where similar results hold. In all these cases, D D D\mathscr{D}D is a Calabi-Yau-2 category: there is a functorial isomorphism Hom ( E , F ) = Hom ( F , E [ 2 ] ) Hom ⁡ ( E , F ) = Hom ⁡ ( F , E [ 2 ] ) ∨ Hom(E,F)=Hom(F,E[2])^(vv)\operatorname{Hom}(E, F)=\operatorname{Hom}(F, E[2])^{\vee}Hom⁡(E,F)=Hom⁡(F,E[2])∨ for all E , F D E , F ∈ D E,F inDE, F \in \mathscr{D}E,F∈D.
Moreover, it has an associated integral weight two Hodge structure H ( D , Z ) H ∗ ( D , Z ) H^(**)(D,Z)H^{*}(\mathscr{D}, \mathbb{Z})H∗(D,Z) with an even pairing ( _ , _ ) _ , _ (_,_)\left(\_, \_\right)(_,_); in the case of a K 3 K 3 K3\mathrm{K} 3K3 surface, H ( D b ( S ) , Z ) = H ( S , Z ) H ∗ D b ( S ) , Z = H ∗ ( S , Z ) H^(**)(D^(b)(S),Z)=H^(**)(S,Z)H^{*}\left(\mathrm{D}^{\mathrm{b}}(S), \mathbb{Z}\right)=H^{*}(S, \mathbb{Z})H∗(Db(S),Z)=H∗(S,Z) with H 0 H 0 H^(0)H^{0}H0 and H 4 H 4 H^(4)H^{4}H4 considered to be ( 1 , 1 ) ( 1 , 1 ) (1,1)(1,1)(1,1)-classes; in the other cases, the underlying lattice is the same, and after the initial indirect construction in [1] there is now an intrinsic construction based on the topological K-theory and Hochschild homology of D D D\mathscr{D}D [64]. There is a Mukai vector v : K 0 ( D ) H 1 , 1 ( D , Z ) v : K 0 ( D ) → H 1 , 1 ( D , Z ) v:K_(0)(D)rarrH^(1,1)(D,Z)v: K_{0}(\mathscr{D}) \rightarrow H^{1,1}(\mathscr{D}, \mathbb{Z})v:K0(D)→H1,1(D,Z) satisfying χ ( E , F ) = ( v ( E ) , v ( F ) ) χ ( E , F ) = − ( v ( E ) , v ( F ) ) chi(E,F)=-(v(E),v(F))\chi(E, F)=-(v(E), v(F))χ(E,F)=−(v(E),v(F)) for all E , F D E , F ∈ D E,F inDE, F \in \mathscr{D}E,F∈D.
In all three cases, there is a main component Stab ( D ) Stab ( D ) Stab † ⁡ ( D ) ⊂ Stab ⁡ ( D ) Stab^(†)(D)sub Stab(D)\operatorname{Stab}^{\dagger}(\mathscr{D}) \subset \operatorname{Stab}(\mathscr{D})Stab†⁡(D)⊂Stab⁡(D) with an effective version of Theorem 2.2 for Λ = H 1 , 1 ( D , Z ) Λ = H 1 , 1 ( D , Z ) Lambda=H^(1,1)(D,Z)\Lambda=H^{1,1}(\mathscr{D}, \mathbb{Z})Λ=H1,1(D,Z) : the map Z Z Z\mathbb{Z}Z is a covering of an explicitly described open subset of Hom ( Λ , C Hom ⁡ ( Λ , C Hom(Lambda,C\operatorname{Hom}(\Lambda, \mathbb{C}Hom⁡(Λ,C ), see [21] for case (1), [8,9] for case (2), and [65] for case (3).
Now consider a family of such K 3 K 3 K3\mathrm{K} 3K3 categories, given by a family of K 3 K 3 K3\mathrm{K} 3K3 surfaces or Fano fourfolds over a base scheme, respectively. In this case, Mukai's classical deformation argument applies: every stable object E E EEE in a given fiber is simple, i.e., it satisfies Hom ( E , E ) = C Hom ⁡ ( E , E ) = C Hom(E,E)=C\operatorname{Hom}(E, E)=\mathbb{C}Hom⁡(E,E)=C, and so Ext 2 ( E , E ) = C Ext 2 ⁡ ( E , E ) = C Ext^(2)(E,E)=C\operatorname{Ext}^{2}(E, E)=\mathbb{C}Ext2⁡(E,E)=C by Serre duality; therefore the obvious obstruction to extending E E EEE across the family, namely that v ( E ) v ( E ) v(E)v(E)v(E) remains a Hodge class, is the only one. Extending such deformation arguments to D D D\mathscr{D}D was the original motivation for introducing stability conditions for families of noncommutative varieties, see [8, SECTION 31]. They allows us to deduce nonemptiness of moduli spaces from the previously known case of K 3 K 3 K3\mathrm{K} 3K3 surfaces (and simplify the previous classical argument for Gieseker stability on K3 surfaces by reduction to elliptically fibered K 3 K 3 K3\mathrm{K} 3K3 s, see [18]), which leads to the following result.
Theorem 2.6 (Mukai, Huybrechts, O'Grady, Yoshioka, Toda [ 8 , 12 , 65 ] [ 8 , 12 , 65 ] [8,12,65][8,12,65][8,12,65] ). Let v H 1 , 1 ( D , Z ) v ∈ H 1 , 1 ( D , Z ) v inH^(1,1)(D,Z)v \in H^{1,1}(\mathscr{D}, \mathbb{Z})v∈H1,1(D,Z) be primitive, and σ Stab ( D ) σ ∈ Stab † ⁡ ( D ) sigma inStab^(†)(D)\sigma \in \operatorname{Stab}^{\dagger}(\mathcal{D})σ∈Stab†⁡(D) be generic. Then M σ ( v ) M σ ( v ) M_(sigma)(v)M_{\sigma}(v)Mσ(v) is nonempty if and only if v 2 := ( v , v ) 2 v 2 := ( v , v ) ≥ − 2 v^(2):=(v,v) >= -2v^{2}:=(v, v) \geq-2v2:=(v,v)≥−2; in this case, it is a smooth projective irreducible holomorphic symplectic (IHS) variety.
More precisely, M σ ( v ) M σ ( v ) M_(sigma)(v)M_{\sigma}(v)Mσ(v) is of K [ n ] K [ n ] K^([n])\mathrm{K}^{[n]}K[n]-type, where n = ( v 2 + 2 ) / 2 n = v 2 + 2 / 2 n=(v^(2)+2)//2n=\left(v^{2}+2\right) / 2n=(v2+2)/2, i.e., it is deformation equivalent to the Hilbert scheme of n n nnn points on a K3 surface (see [25,34] for the basic theory of irreducible holomorphic symplectic varieties). If v 2 2 v 2 ≥ 2 v^(2) >= 2v^{2} \geq 2v2≥2, the Mukai morphism
ϑ : H 2 ( M σ ( v ) , Z ) H ( D , Z ) Ï‘ : H 2 M σ ( v ) , Z → H ∗ ( D , Z ) vartheta:H^(2)(M_(sigma)(v),Z)rarrH^(**)(D,Z)\vartheta: H^{2}\left(M_{\sigma}(v), \mathbb{Z}\right) \rightarrow H^{*}(\mathscr{D}, \mathbb{Z})Ï‘:H2(Mσ(v),Z)→H∗(D,Z)
induced by a (quasi)universal family gives an identification of H 2 ( M σ ( v ) , Z ) H 2 M σ ( v ) , Z H^(2)(M_(sigma)(v),Z)H^{2}\left(M_{\sigma}(v), \mathbb{Z}\right)H2(Mσ(v),Z) with v v ⊥ v^(_|_)v^{\perp}v⊥. If v 2 = 0 v 2 = 0 v^(2)=0v^{2}=0v2=0, then M σ ( v ) M σ ( v ) M_(sigma)(v)M_{\sigma}(v)Mσ(v) is a K 3 K 3 K3\mathrm{K} 3K3 surface and H 2 ( M σ ( v ) , Z ) H 2 M σ ( v ) , Z H^(2)(M_(sigma)(v),Z)H^{2}\left(M_{\sigma}(v), \mathbb{Z}\right)H2(Mσ(v),Z) is identified with v / v v ⊥ / v v^(_|_)//vv^{\perp} / vv⊥/v.
Knowing exactly which semistable objects exist then allows us to describe exactly when we are on a wall. While a complete result as in [11, THEOREM 5.7] also needs to treat essential aspects of the wall-crossing behavior, the basic result is simple to state:
Theorem 2.7 ([11]). Let v H 1 , 1 ( D , Z ) v ∈ H 1 , 1 ( D , Z ) v inH^(1,1)(D,Z)v \in H^{1,1}(\mathscr{D}, \mathbb{Z})v∈H1,1(D,Z) be a primitive class. Then σ = ( Z , P ) Stab ( D ) σ = ( Z , P ) ∈ Stab † ⁡ ( D ) sigma=(Z,P)inStab^(†)(D)\sigma=(Z, \mathcal{P}) \in \operatorname{Stab}^{\dagger}(\mathscr{D})σ=(Z,P)∈Stab†⁡(D) lies on a wall for v v vvv if and only if there exists classes a , b H 1 , 1 ( D , Z ) a , b ∈ H 1 , 1 ( D , Z ) a,b inH^(1,1)(D,Z)a, b \in H^{1,1}(\mathscr{D}, \mathbb{Z})a,b∈H1,1(D,Z) with v = a + b v = a + b v=a+bv=a+bv=a+b, a 2 , b 2 2 a 2 , b 2 ≥ − 2 a^(2),b^(2) >= -2a^{2}, b^{2} \geq-2a2,b2≥−2 and Z ( a ) , Z ( b ) Z ( a ) , Z ( b ) Z(a),Z(b)Z(a), Z(b)Z(a),Z(b) are positive real multiples of Z ( v ) Z ( v ) Z(v)Z(v)Z(v).
And the fundamental reason is similarly simple to explain: by Theorem 2.6, this allows for the existence of extensions
(2.1) 0 A E B 0 (2.1) 0 → A → E → B → 0 {:(2.1)0rarr A rarr E rarr B rarr0:}\begin{equation*} 0 \rightarrow A \rightarrow E \rightarrow B \rightarrow 0 \tag{2.1} \end{equation*}(2.1)0→A→E→B→0
where v ( A ) = a , v ( E ) = v , v ( B ) = b v ( A ) = a , v ( E ) = v , v ( B ) = b v(A)=a,v(E)=v,v(B)=bv(A)=a, v(E)=v, v(B)=bv(A)=a,v(E)=v,v(B)=b, and A , E , B A , E , B A,E,BA, E, BA,E,B are all semistable of the same phase For stronger results, we need to know when such E E EEE can become stable near the wall.

3. CONSTRUCTIONS BASED ON K3 CATEGORIES

In this section we present three applications of stability conditions on K 3 K 3 K3\mathrm{K} 3K3 categories, to irreducible holomorphic symplectic varieties, to curves, and to cubic fourfolds.

3.1. Curves in irreducible holomorphic symplectic manifolds

Let M M MMM be a smooth projective irreducible holomorphic symplectic (IHS) variety of K 3 [ n ] K 3 [ n ] K3^([n])\mathrm{K3}^{[n]}K3[n]-type, with n 2 n ≥ 2 n >= 2n \geq 2n≥2. We let q M q M q_(M)q_{M}qM be the Beauville-Bogomolov-Fujiki quadratic form on H 2 ( M , Z ) H 2 ( M , Z ) H^(2)(M,Z)H^{2}(M, \mathbb{Z})H2(M,Z). By [25, sEction 3.7.1], there exists a canonical extension
ϑ M : ( H 2 ( M , Z ) , q M ) Λ ~ M Ï‘ M : H 2 ( M , Z ) , q M ↪ Λ ~ M vartheta_(M):(H^(2)(M,Z),q_(M))↪ widetilde(Lambda)_(M)\vartheta_{M}:\left(H^{2}(M, \mathbb{Z}), q_{M}\right) \hookrightarrow \widetilde{\Lambda}_{M}Ï‘M:(H2(M,Z),qM)↪Λ~M
of lattices and weight-2 Hodge structures, where the lattice Λ ~ M Λ ~ M widetilde(Lambda)_(M)\widetilde{\Lambda}_{M}Λ~M is isometric to the extended K 3 K 3 K3\mathrm{K} 3K3 lattice U 4 E 8 ( 1 ) 2 U ⊕ 4 ⊕ E 8 ( − 1 ) ⊕ 2 U^(o+4)o+E_(8)(-1)^(o+2)U^{\oplus 4} \oplus E_{8}(-1)^{\oplus 2}U⊕4⊕E8(−1)⊕2. Let us denote by v Λ ~ M v ∈ Λ ~ M v in widetilde(Lambda)_(M)v \in \widetilde{\Lambda}_{M}v∈Λ~M a generator of ϑ ( H 2 ( M , Z ) ) Ï‘ H 2 ( M , Z ) ⊥ vartheta(H^(2)(M,Z))^(_|_)\vartheta\left(H^{2}(M, \mathbb{Z})\right)^{\perp}Ï‘(H2(M,Z))⊥ : it is of type ( 1 , 1 ) ( 1 , 1 ) (1,1)(1,1)(1,1) and square v 2 = 2 n 2 v 2 = 2 n − 2 v^(2)=2n-2v^{2}=2 n-2v2=2n−2. The lattice Λ ~ M Λ ~ M widetilde(Lambda)_(M)\widetilde{\Lambda}_{M}Λ~M is called the Markman-Mukai lattice associated to M M MMM. If M = M σ ( v ) M = M σ ( v ) M=M_(sigma)(v)M=M_{\sigma}(v)M=Mσ(v), for a stability condition σ Stab ( D b ( S ) ) σ ∈ Stab † ⁡ D b ( S ) sigma inStab^(†)(D^(b)(S))\sigma \in \operatorname{Stab}^{\dagger}\left(\mathrm{D}^{\mathrm{b}}(S)\right)σ∈Stab†⁡(Db(S)) on a K 3 K 3 K3\mathrm{K} 3K3 surface S S SSS, then Λ ~ M = H ( S , Z ) Λ ~ M = H ∗ ( S , Z ) widetilde(Lambda)_(M)=H^(**)(S,Z)\widetilde{\Lambda}_{M}=H^{*}(S, \mathbb{Z})Λ~M=H∗(S,Z) with the Mukai pairing, the notation for the vector v v vvv is coherent, and ϑ M Ï‘ M vartheta_(M)\vartheta_{M}Ï‘M is the Mukai morphism mentioned after Theorem 2.6.
We let Pos ( M ) Pos ⁡ ( M ) Pos(M)\operatorname{Pos}(M)Pos⁡(M) be the connected component of the positive cone of M M MMM containing an ample divisor class:
Pos ( M ) := { D H 2 ( M , R ) : q M ( D ) > 0 } + Pos ⁡ ( M ) := D ∈ H 2 ( M , R ) : q M ( D ) > 0 + Pos(M):={D inH^(2)(M,R):q_(M)(D) > 0}^(+)\operatorname{Pos}(M):=\left\{D \in H^{2}(M, \mathbb{R}): q_{M}(D)>0\right\}^{+}Pos⁡(M):={D∈H2(M,R):qM(D)>0}+
The following result rephrases and proves a conjecture by Hassett-Tschinkel and gives a complete description of the ample cone of M M MMM.
Theorem 3.1. Let M M MMM be a smooth projective IHS variety of K [ n ] K [ n ] K^([n])\mathrm{K}^{[n]}K[n]-type. The ample cone of M M MMM is a connected component of
Theorem 3.1 is proved in [11] for moduli spaces of stable sheaves/complexes on a K 3 K 3 K3\mathrm{K} 3K3 surface, and extended in [7] to all IHS of K 3 [ n ] K 3 [ n ] K3^([n])\mathrm{K} 3{ }^{[n]}K3[n]-type, by using deformation theory of rational curves on IHS varieties.
The approach to Theorem 3.1 via wall-crossing is as follows. Let S S SSS be a K3 surface and M = M σ 0 ( v ) M = M σ 0 ( v ) M=M_(sigma_(0))(v)M=M_{\sigma_{0}}(v)M=Mσ0(v) be a moduli space of σ 0 σ 0 sigma_(0)\sigma_{0}σ0-stable objects in D b ( S ) D b ( S ) D^(b)(S)\mathrm{D}^{\mathrm{b}}(S)Db(S), where v v ∈ v inv \inv∈ H 1 , 1 ( D b ( S ) , Z ) H 1 , 1 D b ( S ) , Z H^(1,1)(D^(b)(S),Z)H^{1,1}\left(\mathrm{D}^{\mathrm{b}}(S), \mathbb{Z}\right)H1,1(Db(S),Z) is a primitive vector of square v 2 2 v 2 ≥ 2 v^(2) >= 2v^{2} \geq 2v2≥2. As σ σ sigma\sigmaσ varies in the chamber φ φ varphi\varphiφ containing σ 0 σ 0 sigma_(0)\sigma_{0}σ0, Theorem 2.4 gives a family of ample divisor classes σ â„“ σ â„“_(sigma)\ell_{\sigma}ℓσ in Pos ( M ) Pos ⁡ ( M ) Pos(M)\operatorname{Pos}(M)Pos⁡(M). When σ σ sigma\sigmaσ reaches a wall of â„“ â„“\mathcal{\ell}â„“, as given by Theorem 2.7, the class σ â„“ σ â„“_(sigma)\ell_{\sigma}ℓσ remains nef. On the other hand, consider an object E E EEE that becomes strictly semistable on the wall, admitting an exact
sequence as in (2.1). Varying the extension class in a line in P ( Ext 1 ( B , A ) ) P Ext 1 ⁡ ( B , A ) P(Ext^(1)(B,A))\mathbb{P}\left(\operatorname{Ext}^{1}(B, A)\right)P(Ext1⁡(B,A)) produces a P 1 P 1 P^(1)\mathbb{P}^{1}P1 of such objects, and Theorem 2.4 shows that σ â„“ σ â„“_(sigma)\ell_{\sigma}ℓσ has degree zero on this curve. We have found an extremal curve and, dually, a boundary wall of the ample cone.
FIGURE 1
The approach to Theorem 3.1.
We summarize the history underlying Theorem 3.1 with the diagram in Figure 1. The analogue of Theorem 2.6 for Gieseker-stable sheaves involves a two-step argument, using autoequivalences and deformations. Wall-crossing techniques then imply the existence of Bridgeland stable objects on K3 surfaces, and thus Theorem 2.6. As discussed above, a finer wall-crossing analysis based on Theorem 2.7 then produces the extremal rational curves on moduli spaces that appear implicitly as extremal curves in Theorem 3.1. Finally, another deformation argument, involving rational curves, deduces Theorem 3.1 for all IHS manifolds of K 3 [ n ] K 3 [ n ] K3^([n])\mathrm{K} 3{ }^{[n]}K3[n]-type. Wall-crossing combined with stability conditions in families can also simplify the approach to Theorem 2.6, see [18].

3.2. Curves

Consider a Brill-Noether (BN) wall in Stab ( D b ( X ) ) Stab ⁡ D b ( X ) Stab(D^(b)(X))\operatorname{Stab}\left(\mathrm{D}^{\mathrm{b}}(X)\right)Stab⁡(Db(X)) for a variety X X XXX : the structure sheaf O X O X O_(X)\mathcal{O}_{X}OX is stable and of the same phase ϕ Ï• phi\phiÏ• as objects E E EEE of a fixed class v v vvv. Then O X O X O_(X)\mathcal{O}_{X}OX is an object of the abelian category P ( ϕ ) P ( Ï• ) P(phi)\mathcal{P}(\phi)P(Ï•) with no subobjects; hence the evaluation map O X O X ⊗ O_(X)ox\mathcal{O}_{X} \otimesOX⊗ H 0 ( E ) E H 0 ( E ) → E H^(0)(E)rarr EH^{0}(E) \rightarrow EH0(E)→E must be injective, giving a short exact sequence
(3.1) 0 O X H 0 ( E ) E Q 0 (3.1) 0 → O X ⊗ H 0 ( E ) → E → Q → 0 {:(3.1)0rarrO_(X)oxH^(0)(E)rarr E rarr Q rarr0:}\begin{equation*} 0 \rightarrow \mathcal{O}_{X} \otimes H^{0}(E) \rightarrow E \rightarrow Q \rightarrow 0 \tag{3.1} \end{equation*}(3.1)0→OX⊗H0(E)→E→Q→0
where Q P ( ϕ ) Q ∈ P ( Ï• ) Q inP(phi)Q \in \mathscr{P}(\phi)Q∈P(Ï•) is also semistable. Applying known inequalities for Chern classes of semistable objects to ch ( Q ) = v h 0 ( E ) ch ( O X ) ch ⁡ ( Q ) = v − h 0 ( E ) ch ⁡ O X ch(Q)=v-h^(0)(E)ch(O_(X))\operatorname{ch}(Q)=v-h^{0}(E) \operatorname{ch}\left(\mathcal{O}_{X}\right)ch⁡(Q)=v−h0(E)ch⁡(OX) can directly lead to bounds on h 0 ( E ) h 0 ( E ) h^(0)(E)h^{0}(E)h0(E).
This simple idea turns out to be powerful. For a K 3 K 3 K3\mathrm{K} 3K3 surface S S SSS, we can be more precise: applying Theorem 2.6 to the class of Q Q QQQ, we can construct all E E EEE with given r = h 0 r = h 0 r=h^(0)r=h^{0}r=h0 as a Grassmannian bundle Gr ( r , Ext 1 ( Q , O S ) ) Gr ⁡ r , Ext 1 ⁡ Q , O S Gr(r,Ext^(1)(Q,O_(S)))\operatorname{Gr}\left(r, \operatorname{Ext}^{1}\left(Q, \mathcal{O}_{S}\right)\right)Gr⁡(r,Ext1⁡(Q,OS)) of extensions over the moduli space of such Q Q QQQ.
Corollary 3.2. Let S S SSS be a K 3 K 3 K3K 3K3 surface, v H 1 , 1 ( S , Z ) v ∈ H 1 , 1 ( S , Z ) v inH^(1,1)(S,Z)v \in H^{1,1}(S, \mathbb{Z})v∈H1,1(S,Z) primitive and σ σ sigma\sigmaσ be a stability condition near the Brill-Noether wall for v v vvv. If the lattice generated by v v vvv and v ( O S ) v O S v(O_(S))v\left(\mathcal{O}_{S}\right)v(OS) is saturated, then the locus of objects E M σ ( v ) E ∈ M σ ( v ) E inM_(sigma)(v)E \in M_{\sigma}(v)E∈Mσ(v) with h 0 ( E ) = r h 0 ( E ) = r h^(0)(E)=rh^{0}(E)=rh0(E)=r has expected dimension.
In [5], this is applied, in the case where Pic ( S ) = Z H Pic ⁡ ( S ) = Z ⋅ H Pic(S)=Z*H\operatorname{Pic}(S)=\mathbb{Z} \cdot HPic⁡(S)=Z⋅H, to rank zero classes of the form v = ( 0 , H , s ) v = ( 0 , H , s ) v=(0,H,s)v=(0, H, s)v=(0,H,s). In this case, there are no walls between the B N B N BN\mathrm{BN}BN wall and the large volume limit; hence Corollary 3.2 applies in the large volume limit, and thus to zero-dimensional torsion sheaves supported on curves in the primitive linear system. This gives a variant of Lazarsfeld's proof [49] of the Brill-Noether theorem: every curve in the primitive linear system is Brill-Noether general.
This approach has been significantly strengthened in [27]: instead of requiring E E EEE to be semistable near the Brill-Noether wall, it is sufficient to control the classes occurring in its HN filtration. A bound on h 0 h 0 h^(0)h^{0}h0 is obtained by applying Corollary 3.2 to all HN filtration factors. Thus we need to consider a point near the Brill-Noether walls for all H N H N HN\mathrm{HN}HN factors, and which is the limit point where Z ( O X ) 0 Z O X ∼ 0 Z(O_(X))∼0Z\left(\mathcal{O}_{X}\right) \sim 0Z(OX)∼0.
Proposition 3.3 ([27, PRoposition 3.4]). Let S S SSS be a K3 surface of Picard rank one. There exists a limit point σ ¯ Ïƒ ¯ bar(sigma)\bar{\sigma}σ¯ of the space of stability conditions, with central charge Z ¯ Z ¯ bar(Z)\bar{Z}Z¯, and a constant C C CCC, such that for (most) objects in the heart, we have
h 0 ( E ) + h 1 ( E ) C l | Z ¯ ( E l / E l 1 ) | h 0 ( E ) + h 1 ( E ) ≤ C â‹… ∑ l   Z ¯ E l / E l − 1 h^(0)(E)+h^(1)(E) <= C*sum_(l)|( bar(Z))(E_(l)//E_(l-1))|h^{0}(E)+h^{1}(E) \leq C \cdot \sum_{l}\left|\bar{Z}\left(E_{l} / E_{l-1}\right)\right|h0(E)+h1(E)≤C⋅∑l|Z¯(El/El−1)|
where E 0 E 1 E m E 0 ⊂ E 1 ⊂ ⋯ ⊂ E m E_(0)subE_(1)sub cdots subE_(m)E_{0} \subset E_{1} \subset \cdots \subset E_{m}E0⊂E1⊂⋯⊂Em is the H N H N HNH NHN filtration of E E EEE near σ ¯ Ïƒ ¯ bar(sigma)\bar{\sigma}σ¯.
The following application completes a program originally proposed by Mukai [62]:
Theorem 3.4 ([27, 28]). Let S S SSS be a polarised K 3 K 3 K3K 3K3 surface with Pic ( S ) = Z H Pic ⁡ ( S ) = Z ⋅ H Pic(S)=Z*H\operatorname{Pic}(S)=\mathbb{Z} \cdot HPic⁡(S)=Z⋅H and genus g 11 g ≥ 11 g >= 11g \geq 11g≥11, and let C | H | C ∈ | H | C in|H|C \in|H|C∈|H|. Then S S SSS is the unique K3 surface containing C C CCC, and can be reconstructed as a Fourier-Mukai partner of a Brill-Noether locus of stable vector bundles on C C CCC with prescribed number of sections.
The structure of the argument is as follows. The numerics are chosen such that there is a two-dimensional moduli space S ^ S ^ hat(S)\hat{S}S^, necessarily a K3 surface, of stable bundles E E EEE on S S SSS whose restriction E | C E C E|_(C)\left.E\right|_{C}E|C is automatically in the Brill-Noether locus. Conversely, given a stable bundle V V VVV on C C CCC, its push-forward i V i ∗ V i_(**)Vi_{*} Vi∗V along i : C S i : C ↪ S i:C↪Si: C \hookrightarrow Si:C↪S is stable at the large volume limit. Standard wall-crossing arguments bound its HN filtration near the limit point σ ¯ Ïƒ ¯ bar(sigma)\bar{\sigma}σ¯ in Proposition 3.3, which then gives a bound on h 0 ( V ) h 0 ( V ) h^(0)(V)h^{0}(V)h0(V). The argument also shows that equality-the Brill-Noether condition-only holds for the H N H N HN\mathrm{HN}HN filtration E E | C = i V E ( H ) [ 1 ] E → E C = i ∗ V → E ( − H ) [ 1 ] E rarr E|_(C)=i_(**)V rarr E(-H)[1]\left.E \rightarrow E\right|_{C}=i_{*} V \rightarrow E(-H)[1]E→E|C=i∗V→E(−H)[1], i.e., when V V VVV is the restriction of a vector bundle in S ^ S ^ hat(S)\hat{S}S^. Thus S ^ S ^ hat(S)\hat{S}S^ is a Brill-Noether locus on C C CCC, and S S SSS can be reconstructed as a Fourier-Mukai partner of S ^ S ^ hat(S)\hat{S}S^.

3.3. Surfaces in cubic fourfolds

Let Y P 5 Y ⊂ P 5 Y subP^(5)Y \subset \mathbb{P}^{5}Y⊂P5 be a complex smooth cubic fourfold and let h h hhh be the class of a hyperplane section. Following [38], we say that Y Y YYY is special of discriminant d d ddd, and write Y C d Y ∈ C d Y inC_(d)Y \in \mathscr{C}_{d}Y∈Cd, if there exists a surface Σ Y Σ ⊂ Y Sigma sub Y\Sigma \subset YΣ⊂Y such that h 2 h 2 h^(2)h^{2}h2 and Σ Î£ Sigma\SigmaΣ span a saturated rank two lattice in H 4 ( Y , Z ) H 4 ( Y , Z ) H^(4)(Y,Z)H^{4}(Y, \mathbb{Z})H4(Y,Z) with
det ( h 4 h 2 Σ h 2 Σ Σ 2 ) = d det ⁡ h 4 h 2 â‹… Σ h 2 â‹… Σ Σ 2 = d det([h^(4),h^(2)*Sigma],[h^(2)*Sigma,Sigma^(2)])=d\operatorname{det}\left(\begin{array}{cc} h^{4} & h^{2} \cdot \Sigma \\ h^{2} \cdot \Sigma & \Sigma^{2} \end{array}\right)=d